The Conrad is kind of on the line between the general Battery Park/Financial District area and Tribeca, on the west side. Seamore's across the street (sort of, just ask) always has a good bar scene when I walk by, although I've never walked in - just lots of people. There's Oculus Beer Garden just north of the Oculus. The bar at The Beekman Hotels Temple Court is absolutely gorgeous, and sometimes a good scene (hard to say with Covid, I was there a few weeks ago and it was not very crowded, I think crowds are slowly returning). Locande Verde does have a bar scene, as does Frenchette. There's also a bar near Duane St Park called Weather Up that's fun.
City Harvest might be fun for you, on the water, a little further up at Pier 26, very walkable from The Conrad.
Walk across from west to east on Fulton St and that will put you at the Seaport, there is a big open bar in the middle that has a happy hour scene, and R17 on the rooftop of Pier 17 is nice.
Felice which is the restaurant spilling off of Gild Hall has a happy hour bar scene that sometime re-energizes later in the evening. Get a table in the bar area or eat at the bar - it's very social. Crown Shy is more upscale and also has a good bar scene. They are both in the other direction, further south/east from The Conrad. Also check out the bar at American Cut.