Not specifically for Stamford, but there's a sight for Fairfield County, Stamford being one of the areas: https://fairfield.apaleagues.com/Default.aspx

Nine Ball Cafe and Rack N Roll were the same place, just under different ownership. Unfortunately it's been closed for years. Your best bet is the place the other commenter mentioned in White Plains.

If you really want Stamford, the only place I'm aware of that has more than 2 tables and isn't a private club is El Rocoto. It's a restaurant attached to the motel off exit 9. 3 tables, cheaper than some other bars, and pretty good food. Down the street from them is La Vista Bistro. They only have 2 tables, but they're also a little cheaper than some other bars. My last suggestion would be Twin Rinks. There's a bar upstairs with 2 tables that they keep in really good condition and there's not usually people playing on them.

If you have any questions you're welcome to DM me. I play in the Stamford 8 ball league so I know the pool options.