I encountered the same problem with my g'daughter's t'shirt quilt. I used the back side of the same shirt and cut pieces to sew to the logo piece to bring it to the right size. Worked beautifully.
Also had the same problem of too many white t-shirts but I used a black fabric with red, yellow, blue, etc. small dots on it and it was perfect to set off all the white and light colored shirts. Put a bring yellow on the back. It turned out to be the best looking t-shirt quilt I've made yet and I've made one for all my grandchildren when they graduated from hi school.
I also used a lot of the small logos on the pockets and worked them into the outer sashing strip. Then I machine embroidered their name and graduation date in the sashing strip corner.
Good luck.