Hey there Cayman! New Kingston is the commercial hub for Jamaica...it's also Kingston's "Hip Strip". You can meet nice Jamaican girls just about anywhere - if she asks you to buy her a drink - that's a sign of a potential hustler!
So...if you want an eclectic crowd, then Quad is the place - it's four night clubs in one, each with different themes and ofcourse a different vibe. However, if you want to really mingle with locals, Asylum is just around the block on Knutsford Boulevard - this club tends to be a little more hardcore and caters to mainly young adults and is very popular on weekends. Beside the Asylum is Jamrock Cafe, a very nice place to mingle and get some good food at the same time. Similar to this one is Cuddy's, located at the New Kingston Shopping Centre - it's a new sports bar run by a famous Jamaica/West Indies cricketer (Courtney Walsh)...very laid back spot with a techy flair, in that, there are several booths for eating, each booth having a computer/TV (including local entertainment channels) plus flats screens all over showing the various sports channels. At the entrance/exit (not sure) there is a mini stadium with a big screen setup for major sport events (NFL, NBA, Boxing etc). You'll find mostly groups here so not sure how meeting single women would go...
Now for the really laid back "I just want to chill with you" atmosphere, you can visit the park opposite The Pegasus (Emancipation Park). Alot of women go there to walk/jog/relax in the evenings (it sure would be alot quieter here). Adjacent to the park is a miniature golf spot and a mini driving range/golf academy (not sure if you would be interested in these types) but who knows...maybe you can do some putt-ing tricks..lol.
Kingston buzzes with all types of women, the hustlers/hoes come out at night usually. While in New Kingston, you can visit the tourist board offices and get more information on where to go/what to do. Hope this helps. Cheers!