Yes. They can put a request in to have PSA log it as a new variation, but PSA normally won't unless it the set has been fully checklisted with the variations in a card catalog or price guide like Beckett or the Standard Catalog. So really, the first step should be to get it officially checklisted. Once that's done, you've got to work with PSA to have them declare it a variation they check for.
I have a couple that I've had graded hoping to get them checklisted over time, then resend in for reslabbing once they've made that variation:
1972 Topps McNertney with yellow fingerprint
1963 Topps Bob Veale with blue and purple streak in sky
1960 Topps Master & Mentor Mays/Rigney with extra brown box on back
There are also two 1967 Laughlin Mickey Mantle cards (one with black spot on pants)
I think all these are in the Show me your print defects thread.