Yorkshire Terrier was (in later years) part of Yorkshire Traction, so now part of Stagecoach.
The next biggest player in town is TM Travel, part of Wellglade (Trent Barton etc), who have picked up a few titbits from First in recent years (the depot at Halfway, 30 from Sheffield to Crystal Peaks, 6 from Darnall to Millhouses etc) rather than actually competed with them.
TM Travel's lower cost base and dozens of Solos would be more suited to many of the residual Rotherham routes (the ones that don't run through to Doncaster or Sheffield).
Stagecoach are the only operator to have a Rotherham base (Rawmarsh) but their remaining South Yorkshire depots have taken on extra work after Doncaster and Shafton closed (with Stagecoach shifting routes between depots to try to balance things), so I don't know if they'd have space. The First 39/ 40/ 41/ 42 run in that general direction).
http://www.travelsouthyorkshire.com/rotherhambusnetworkmap/
I just don't know if the remaining services are really worth any operator taking on.
Whilst Stagecoach (and Yorkshire Traction/ Terrier before them) have competed on a number of corridors in Sheffield, there's not been competition on most Rotherham services.
Yorkshire Terrier did compete on the old 137/138 corridor twenty-plus years ago, and Tates Travel used to compete on the East Dene services (56/57 vs First's 11), but most of the competition has been on "out of town" routes (e.g. First's 22 to the Dearne Valley and Barnsley competing with existing Stagecoach services).
Plus, with the TramTrain opening in the next few years (who knows? keeps getting pushed back...) that will suck demand from the Sheffield - Meadowhall South - Rotherham - Parkgate corridor, which might make things even bleaker, as far as commercial bus operations go.
Sorry, probably a longer answer than you were wanting, but I don't think Rotherham is going to be fertile territory for any operator. If First had pulled out a couple of years ago, I could have seen Tates Travel make a play, but we are generally without those kind of cowboys now.