Wilder had spent six years in charge at Halifax, but left the club in June after they were demoted three divisions from the Blue Square Premier to Unibond Division One North after a long struggle with financial problems.
Smith's final game in temporary charge was Saturday's 1-1 draw with Stevenage at the Kassam Stadium, which left the U's in 13th place in the Blue Square Premier - 11 points behind the play-off places - going into the festive fixture programme.
Wilder, meanwhile, had been with Bury for the final time as they lost 2-0 at Bournemouth.
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