Finnigan, 32, picked up the knock in January's win at Witton Albion and was initially expected to be out for around a month, but it has proved worst than first feared following further tests and scans.
He has scored 14 goals in 22 appearances this term, and Shields say that no timescale has been set for a return as yet.
South Shields travel to Grantham Town in the EVO-STIK League Premier Division on Tuesday night knowing that a win can put them back top of the league ahead of the visit of current leaders Warrington Town on Saturday.
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