My experience with the libraries that Oppert listed is that one has to actually go to the locations and ask local families if any of the manuscripts are still extant.  Some may have been folded into the TMSSML in Thanjavur, but you have to go there too, in practice, to see any manuscripts.  Copies of MSS in the TMSSML are only given with the permission of the District Collector, usually a busy, non-academic political appointee.  There can be a bit of bias too about Sanskrit (rather than Tamil) materials, so diplomacy is advised.  Some family collections may also have been absorbed by other major cultural centres such as Kumbakonam, though I don't have any info on that.

See some of the papers in