TRACON Controller described a below MVA event apparent caused by the Tower Controller failing to climb the departure after being issued 2000 prior to visual separation applications.

Narrative

Aircraft X checked on at 2000; said they were told to maintain 2000 for now. They should had been climbing to 8000. The lowest MVA in the area is 3600. The minimum altitude for the DP they were on (Jackson 8) is 7000. SUPPOSITION: ZZZ1 Tower was providing visual separation between Aircraft X and Aircraft Y; an arrival on a straight in Visual Approach to Runway XX. The Tower called and advised that Aircraft Y was circling for right traffic to Runway XX. Aircraft Y appeared to over fly the runway and entered left traffic Runway XX. Aircraft X was departing Runway XX right below them. Apparently; the tower restricted Aircraft X to 2000 and forgot to reissue 8000 before switching them to departure. Supposition aside; since I don't know the factual cause of the event I can't comment on how to fix it. However; if this was caused by insufficient spacing between the arrival and departure; the fact that the shout line is OTS may have contributed since it takes longer to coordinate with the dial lines. In that case; fixing the shout line would help.