One option for the end of Mass on the Solemnity of the Holy Trinty is the Te Deum that epic hymn of praise. Sometimes this is a “Solemn Te Deum,” with two thurifers on either of the altar swinging their thurifers and lots of bell ringing.
Wikipedia on the uses of the Te Deum
In the traditional Office, the Te Deum is sung at the end of Matins on all days when the Gloria is said at Mass; those days are all Sundays outside Advent, Septuagesima, Lent, and Passiontide; on all feasts (except the Triduum) and on all ferias during Eastertide. Before the 1962 reforms, neither the Gloria nor the Te Deum were said on the feast of the Holy Innocents, unless it fell on Sunday, as they were martyred before the death of Christ and therefore could not immediately attain the beatific vision.
In the Liturgy of the Hours of Paul VI, the Te Deum is sung at the end of the Office of Readings on days when the Gloria is sung (Sundays outside Advent and Lent), and all solemnities, including the octaves of Easter and Christmas, and all feasts) and also on the Sundays of Advent. It is also used together with the standard canticles in Morning Prayer as prescribed in the Anglican Book of Common Prayer, in Matins for Lutherans, and is retained by many other churches of the Reformed tradition. It is also used by the
Eastern Orthodox Churches in the Paraklesis (Moleben) of Thanksgiving.
There is a good article on the history of the Te Deum and its structure over at Zenit:
The first Te Deum I ever sang was Mozart’s exuberant piece. In those days I had no idea what I was singing about and we certainly would not have sung it at Mass (Apologies – Catholic education.)
Warning – The Te Deum might be the ultimate hymn of praise to God but it is absolutely exhausting to sing. The Solemn tone is obviously more tiring than the Simple tone. I would advocate a fitness programme in the run up to when you have to sing it- star jumps, power walks etc…