None of the hymns in the “four-hymn sandwich” are required, so the “final song” is just the last one to be sung, which will depend upon which options are employed.
The GIRM does allow for a song between Communion and the Prayer/Blessing/Dismissal:
88. When the distribution of Communion is over, if appropriate, the Priest and faithful pray quietly for some time. If desired, a Psalm or other canticle of praise or a hymn may also be sung by the whole congregation.
- After this, the Priest may return to the chair. A sacred silence may now be observed for some time, or a Psalm or other canticle of praise or a hymn may be sung (cf. no. 88).
There is no call, on the other hand, for any music to follow the dismissal, although one may interpret the similarity to the Entrance Procession to suggest a need to be accompanied by some sort of music.
186. Then, together with the Priest, the Deacon venerates the altar with a kiss, makes a profound bow, and withdraws in a manner similar to the Entrance Procession. [this similarity is not noted, for whatever reason, in instructions for Mass without a deacon]
So all told, it sounds to me like the practice at your parish is certainly atypical for the US, but violates no rule either by placing a song where it does or by omitting the use of a “final song” to cover the procession out.