** Regina Coeli (either chant or polyphonic) for either Sunday
** Confirma Hoc (Handl) or Hodie Completi Sunt (Gabrieli) for Pentecost Sunday
** Cantate Domino (polyphonic - various) for either Sunday
** Veni Creator (either chant or polyphonic) for Pentecost Sunday

(IF you are looking specifically for chant, all of the above are derived from chant, so you could certainly do that setting rather than the polyphonic settings if desired.)

Hope this helps!

These are just helpful reference points:

http://www3.cpdl.org/wiki/index.php/Category:Ascension

http://www3.cpdl.org/wiki/index.php/Category:Pentecost

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For polyphony for this last part of Paschaltide, I am very fond of Byrd's Non Vos Relinquam Orphanos (technically, proper to the Sunday between Ascension Thursday and Pentecost Sunday):

https://www.youtube.com/watch?v=LZ7FXcIimis

"Regina Coeli" by Gregor Aichinger at the very end (after final benediction)

for Pentecost, "Factus est repente" by the same, with the secunda part (confirma hoc). I often use the first part during offertory and the second (quieter) during communion.

"Cantate Domino" by Handel (with french text - specific for Ascension and Pentecôte

"Veni Sancte Spiritus" by Aloys Kunc

"Veni Creator Spiritus" by MR DeLalande

And we use many french hymns, but of little interest for you !
Scores on demand !

Re Offertory of Ascension – the OF books give two chants: Ascendit Deus (as in EF), and Viri Galilei “ad libitum”. In an EF context, one could certainly sing Ascendit Deus as the proper offertory, and then follow it with Viri Galilei (as extra music to fill time).