Quite so.

It also belongs to Sept. 14th (the Exaltation of the Holy Cross), and formerly May 3rd as well (the Finding of the Holy Cross).

For these, note that as it is not Passiontide, there are variants to the line in vs. 6 referring to "Hoc Passionis tempore".

By the way, the monastic variant of this hymn shows up in the (recommended) film The War of the Vendée, as apparently the Vendeans actually did use this hymn.

Vexilla Regis would be fine, you could even sing the original words and verses... see below, I have a copy with the music but it needs corrections!

A couple of other options,
https://societyofstbede.wordpress.com/2014/09/14/feast-of-the-exaltation-of-the-holy-cross/

Vexilla regis prodeunt,
fulget crucis mysterium,
quo carne carnis conditor
suspensus est patibulo.

[Confixa clavis viscera,
tendens manus, vestigia,
redemptionis gratia
hic immolata est hostia.]

Quo vulneratus insuper
mucrone diro lanceae,
ut nos lavaret crimine,
manavit unda, sanguine.

Impleta sunt quae concinit
David fidelis carmine,
dicens: `In nationibus
regnavit a ligno Deus.'

Arbor decora et fulgida,
ornata regis purpura,
electa digno stipite
tam sancta membra tangere.

Beata, cuius brachiis
pretium pependit saeculi,
statera facta est corporis,
praedam tulitque Tartari.

[Fundis aroma cortice,
vincis sapore nectare,
iucunda fructu fertili
plaudis triumpho nobili.

Salve ara, salve victima
de passionis gloria,
qua vita mortem pertulit
et morte vitam reddidit.]

O crux, ave, spes unica
hoc passionis tempore,
auge piis iustitiam
reisque dona veniam.

Te, summa Deus Trinitas,
collaudet omnis spiritus:
quos per crucis mysterium
salvas, rege per saecula.

We are singing Vexilla Regis for the EF Passion Sunday Mass. We are also singing these motets, which can be sung again on Palm Sunday and Good Friday in either form:

Christus Factus Est, Felice Anerio

Christ became obedient for us unto death, even to the death, death on the cross. Therefore God exalted Him and gave Him a name which is above all names (Gradual and Gospel Verse for Palm Sunday and Good Friday)

http://www1.cpdl.org/wiki/images/sheet/aner-chr.pdf
http://www1.cpdl.org/wiki/images/sound/aner-chr.mid
https://www.youtube.com/watch?v=ERsHtf0EAW4 (accidentals here don’t match this score)

O Vos Omnes, Tomas Luís de Victoria – from Tenebrae Responsories (Lamentations):
O all ye that travel along the highway, attend and see if there be any sorrow like unto my sorrow.

https://www.youtube.com/watch?v=m11B9GuDUmM
http://www.kingcharlessingers.org.uk/editions/victoria_o_vos_omnes.pdf

Ah if only my schola were large enough to handle 4 part polyphony! We're doing the chant this year. I hope within another year, I will have them singing solfege confidently enough, and have recruited a few more people to sing this beautiful arrangements.

Where did you get that?

In fact, here's another one. I'm not supposed to say how I got the original.

our schola also want to try Palestrina's Vexilla
http://www.ccwatershed.org/vexilla/

our active member is only about 5 people, with only 2 people can read music with me, who can read music, but have often terrible out of tunes pitch :p

other impromptu member only can show up in mass only. And we using messenger to communicate, so they will practice with practice track from ccwatershed.

this will be adventurous practice .

It's depend..

Our schola, serve mass about only each month .
With 2 times practice, we can sing all proprium .
So in the spare time, we try to push boundary make it better singing technic .
And not me who direct schola, but to train ourself better, make it more challenging, more younger member feel more enthusiast .