Tips for gluing new foam into wing dolly

The foam inside my wing dolly has completely worn out and needs
replacement. I have the replacement foam and have removed all the old
foam and contact cement. The trick will be getting the new felt glued
into position. With contact cement, as soon as the two parts touch,
they are bonded.

Anyone have any pointers or suggestions as to how to best go about
getting the new foam glued into the dolly properly without making a
total mess of the thing?

Thanks for the help!

WD