Only at the entrance with an improvised interlude or fanfare on the hymn tune if you're running out of hymn verses. Done right, you can use the last phrase of the hymn verse to lead into the final verse.

At communion I get a small group to sing the communion chant usually from the SEP until most of the congregation have had communion then I use a hymn.

"Alternatim" strictly speaking is the replacement of alternate verses, either by a different singing ensemble or instrument(s). Because half of a Psalm would not be heard aloud organ verses have at various times been discouraged or forbidden. There's no such reason, on the other hand, for interludes to be addressed by legislation. If you have a choir leading, the congregation knows when to join; if not, they'll still easily get the hang of waiting for the organ to 'breathe' them. What's special about Catholic settings? We do this all the time when the line is long.

At the cathedral, when incense is used (which is nearly every week), the organist often adds improvised interludes before the final verse of the entrance hymn so that the hymn ends as the celebrant reaches the chair.

However, it's never just the straight hymn tune, because this would be confusing, musically, to the congregation.

He rarely uses the last phrase of the hymn to lead back, but also, his playing is so clear that it is obvious when it's time to return.
YMMV, especially if your congregation is not used to it.

I see no problem with it.

We do the Communion Antiphon (in English or Latin) with many verses.
Then there's a long organ improvisation on that mode.
Then we do Psalm 34 to that mode in English with all the verses of the Psalm.
Then there's another long organ improvisation on that mode.
Then we do a motet or hymn.

I have played interludes within hymns countless of times. Picking up on Kathy's rec, I often create extending matter beginning at the last note sung at the end of a verse. Having had a flugelhorn on a weekly basis for years, I sometimes have him play a verse with ornamentation even modulating to a different key (perhaps a half or whole step higher) and then have the congregation return in that new key for the final verse.

Extending matter can be as simple as a repeating motif or fragment of the hymn tune. If it is a recessional hymn, a short two or four measure "fanfare like statement" can signal the congregation that we are raising the melody (can add an air of majestic finality) or even descend a half step during a lenten hymn (can bring a sense of seriousness and resolution to following the way of The Cross).

There are many creative ways to insert instrumental passages. When performing a long chant piece as a hymn, I would sometimes insert a repeat of cadence (last two or four measures of the end of the verse) in improvised instrumental fashion, which gives the congregation a moment to reflect upon the sentence they just completed before moving on to the next verse. I might do this after verses 3, 6 and 9 for instance.

Sometimes I improvise an introductory phrase, such as for "We Three Kings" using a reed playing an Oriental scale (i.e., major scale with flatted 2nd and 6th).

The options are endless and I always try to give deference to the text and let it form the development of musical ideas keeping in mind that we are not performing for performance sake but helping to give "musical flesh" to the theological essence of a particular liturgy. I am always careful and reserved when implementing these ideas; If not executed in tasteful, artistic fashion being sensitive to the Holy Spirit it can turn into a music for musics sake display of something that is tacky, says "look at me", and detracts from the liturgy rather than heightening the experience.

Well, JS Bach used to do things like that, but the
PIPs didn't appreciate it!

Yeah, our PIPs would sing along anyway.

I really don't understand the problem. In Lent, the organ should only be used to support the singing, but surely this is not a rule at other times of the year. Music is a good in itself. The organ is the preferred instrument of the liturgy. What is the problem?

Nothing, there is no difference... I don't understand the problem either. If the organist is competent, and it works well for the situation (for example, long processions), go ahead and do it. If it doesn't, there's obviously no obligation to do so.

Not that much, Adam; the badness only increases logarithmically: therefore 7.2x as bad.

:-)

I like interludes, but I agree with ClergetKubisz that many people might keep singing. Maybe not if there's a noticeable change in registration.

Exactly, Kathy. Even without any registration changes: if you go into the improvisation during the last note of the hymn (instead of after it), I doubt anyone would be confused and think it's time to sing.

PIPs may or may not like hymn interludes. Whatever. But to say it's confusing on where to sing and not to sing? Never, if it's done reasonably well.

1) Start improvising during the last note of the hymn, so there's no ambiguity that the hymn is not just moving on normally.
2) Go as long as you want, w/ or w/o key changes and registration changes. Do whatever the heck you want.
3) At the end, of the improvisation, make it very clear when the hymn is coming back with your chord progressions and tempo, or even improvise into the standard harmonization for the last 4 bars of the hymn, if you wish.
4) Play the next verse of the hymn after the standard pause.

I challenge anyone to try this and say people are confused on when they should be singing.

Sorry, I trusted a flawed tool on the internet. That should be 4.58 = log (2) of 24; even less badness than estimated!

I'm still not sure what the purpose would be of improvising in the middle of a Communion antiphon, or between the verses and the antiphon, or between the verses of a hymn at this time.

As a rule, at my current parish we always sing all the verses of whatever hymn we sing at communion. I occasionally add interludes to communion hymns for two reasons:

(1) The hymn is really long and I want a give people a break.
(2) It looks like the hymn is going to end sooner than I want it to end.

Here's a funny story about interludes: As an organ student at CUA, I was assistant music director at a parish with an extremely short aisle -- therefore interludes for processions were rare and I was not accustomed to playing them. I was asked to play for a big university Mass in the upper church of the Basilica, and the entrance procession, besides the standard ministerial entourage, included dozens of priest and the whole university faculty. You may know that the aisle is pretty darn long, too. Anyway, with the combination of nerves and inexperience, I played all the way through the second-to-last stanza of the hymn (don't remember the text, but it was LASST UNS ERFREUEN) without paying attention to the status of the procession through the cameras. Well, it looked like they had barely begun. So, panicking, I played probably a five-minute interlude before playing the last stanza, and then another minute or two of a fanfare at the end while everyone got into place. We had a good laugh about it afterward.

I like to sing as much as possible during a service and I've been unhappy when the choir is allowed to replace a hymn normally sung by the congregation. I'm not opposed to having the choir sing additional pieces, such as starting a communion with as many hymns as it takes to have everyone back from communion, then allowing the congregation to sing another while the cleanup is being done. That choir piece could also be an instrumental, and I've usually seen it done very quietly and when the congregation's turn to sing, the organist will increase the volume for maybe one long chord as a signal to start singing.