If you sing a hymn for the exit, you should sing the whole thing. Most good solid traditional hymns have 3 to 6 verses, and form a coherent unit. Most of them are not too long to sing, assuming you don't drag them out at dirge-tempo.
If you don't have time and patience for 3 to 6 verses of a good solid hymn, then just play an organ piece.
The Roman Rite provides no recessional. There must be a reason for that. O the one hand, there seems to be a desire for something celebratory as the conclusion of the Mass; this seems to have led the Episcopal tradition to place the Gloria at the end of the Mass in their older tradition. But even they have put the Gloria back at the beginning of the Mass. It has seemed to me that there is a reason for no recessional: the Mass has led to the reception of Communion, after which there is a closing collect, blessing and dismissal. The congregation is left having just received the Lord and ideally will remain in prayer for a while. Recessionals tend to close this off, after which the people simply leave the church. I have sometimes thought that some organ pieces played as recessionals are calculated to drive the people out of the church. Rather, it is appropriate to have some sort of processional music to conduct the ministers out, but after that, the music should be conducive to personal prayer and thanksgiving. I play a short, rhythmic piece followed by a more meditative piece conducive to prayer
I wonder if there is an historical reason, in monastic and collegiate use, for the lack of a recessional at Mass that just filtered down to everyone else. The canons/monks would have sung Terce before the Conventual Missa Solemnis, would they not have simply remained in their stalls to sing Sext?
The EF Caeremoniale Episcoporum asks for organ music for the processional and recessional of the bishop. Quotiescumque Episcopus solemniter celebraturus aut Missae solemni per alium celebrandae in festis solemnioribus interfuturus Ecclesiam ingreditur; aut, re divina peracta, discedit, convenit pulsari organum. Caer. Ep., lib. 1, cap. xxviii, nr. 3. And organ music for the recessional is also the norm for other solemn masses. In missa solemni pulsatur [...] ac in fine missae. Caer. Ep., lib. 1, cap. xxviii, nr. 9.
When short Introits are needed, check out "New Collection: Responsorial Introits". These are by Aristotle Esguerra, and posted in November of 2013. Very good.
To participate in the discussions on Catholic church music, sign in or register as a forum member, The forum is a project of the Church Music Association of America.