To echo what has been said above, Gen Con runs 24/7 for the whole con. Yes, the exhibit hall is only open from 10 to 6, but there are events all the rest of the time. (granted 1am to 8am is probably a light period). There are many "entertainment" events / shows at night - burlesque, comedy, puppets, videos and films. Gen Con Dance is Saturday at the Union Station.
Like games from a specific company - check their websites - may game companies will do events in the evening. For several years Steve Jackson took over a bar and turned it in to Muchkin Tavern. Colts Tavern has also hosted many company events.
Follow a gaming podcast - they may be having a meet up - last several years the Secret Cabal gaming podcast has had meet ups in a hotel bar. Also several of the podcasts do live recorded shows which are official events (Dice Tower, D6 Generation).
Bars - every bar downtown will be full go gamers. Scotty's and the Ram are the best known - but any bar downtown - you will find gamers to hand with.
Beer Garden - last several years Sun King brewery has produced officail Gen Con beer and had a beer garden on Gerogia Street co located with food trucks. Beer and good food truck food - a nice combo.
BEER!!! Beer itself is a topic by itself. Indy (like much of the country) has seen a craft brew explosion. Most is only available in the area. Every bar downtown (even in the hotels) has some selection of local craft beer available. Several have a very wide selection. Sun King, Tow Yard, and Two Deep are close by (Tow Yard is an easy walk) the other two are farther. Tomlinson Tap Room in the Indy City Market is all Indy craft beer and has hosted gaming group meet ups in the past. My guess is that there will be 3 beers brewed specially for the con Sun King (official), Ram, and Scotty's.