I just happen to be listening to the Mawby Ave Verum right now and think it would be very appropriate for a motet after the Communio.

We have used Sacerdotes Domine by William Byrd for a priest's first Holy Mass.
Also, Father Samuel Weber has Propers for the Ordination of a priest on his
website. Would this help out for a first Holy Mass also?

"Dixit Dominus" from Mozart's Solemn Vespers of a Confessor would make a splendid Offertory motet. It includes the words "Tu es sacerdos in aeternam" (You are a priest forever).

Here's a faux bourdon setting of "Psalm for a First Mass".
TTBB setting followed by SATB

When Cardinal O'Connell was Rector of the American College in Rome, he composed the attached hymn, Juravit, to be sung at the first Mass of priests. Later, when he was appointed Archbishop of Boston, Juravit became "standard" at ordinations, first Masses, anniversary Masses, and related occasions. It is still used at times in the Boston Archdiocese for anniversary Masses of priests. Perhaps you will find it useful.

Having been in precisely your position for many friends who were newly ordained ( and very honored to do their music as well) to a man, they all loved the Proulx setting of Gustav Holst's "O God Beyond All Praising" as a recessional. It's both moving, grand and reverent. But don't take it at the suggested tempo of 70 ? (I think) It needs to be done at 80 - 82 (moderato) or else it transforms from lean and majestically powerful to sluggish and overweight.

And, I attended another friends first mass at the Basilica of the National Shrine when Leo Nestor was the Music Director there and was blown away by the opening hymn "Only Begotten, Word of God Eternal" which is almost dirge-like, powerful and otherworldly.
It's in Worship II and III and can be done effectively with the simple SATB parts in the hymnal. The organist needs to come up with a bit of a fanfare before beginning.

Forgive my indulgence, but perhaps this story will be helpful:
(When I witnessed this grand procession - the priests and servers were all male, in cassock and surplus and there were about 10 -15 in the procession. It was in the crypt church below the main church in the Basilica and they formed outside the church with the doors closed. They prayed and were blessed by the main celebrant and incensed before entering the crypt church. As the entrance hymn began, low, rumbling and quiet - they opened the doors and the incense came pouring into the church - then the organist began a powerful dynamic ascent with the pedals and proceeded into the fanfare which was short, but also powerful. As soon as the choir/congregation began singing, the procession entered the church...slowly - in true procession. The hymn grew in dynamics perfectly in line with the text - as the procession appeared through the incense. It was quite something, and I thought I was going to faint because it was so beautiful and powerful. When we got to "hear our petitions, as we throng they temple" the procession and the music became one - and as they made their way through the incense into the church the entire congregation was swept up into the glory of God. It seemed that heaven and earth had indeed become one at that moment and we were given a glimpse of the heavenly liturgy throughout the mass. I just had to tell that story. I'll never forget it. My oldest son was in the procession, as the thurifer...a little boy leading the procession. Incredibly glorious. sigh)

Never heard of that!

The Proulx is an outstanding concertato, TBS, but depending upon the circumstances and disposition of the liturgical ministers and congregation, I'd think about using it at the so-called "hymn of praise" after the Communio. The introduction and interludes of the Proulx, along with the missing 2nd verse (if an OCP parish) for the PIPs is a bit risky if the new "Father" is winding his way down the aisle when sung. The Proulx is a perfect example of sequential, systematic adornment of a worthy hymn, and I'd sure hate to have folks yakking during the third verse when the sopranos (if they have 'em) soar on that beautiful descant. End of caveat.
And yes, you can tell Leo my real name.