To answer your question - Yes you can
I once release a player w/o renewing his contract and later noted him listed in auction as (F)
However it doesn't make sense to sign him again.
The reason is that renewing his contract cost TE $ but bidding for him (though he is a free agent) cost Tokens.
Unless you can equate US$ 1 = ......TE $
1 T = ...... US$ ; and see if it worthwhile to bid again.