It's instructive to know where the power you develop is lost when you're riding at steady-state.
Most of it is lost to air drag against your body.
After that, probably bike frame and wheel drag. That's where a narrower tire has some potential to "faster" you.
Finally, mechanical losses. This is drivetrain friction and rolling resistance from tires. This is where a tire with a better carcass has some potential to "faster" you.
You might see some improvement in speed at low speeds if you go to a different tire, especially if you have one that rolls like ass. Some stock tires for hybrids are pretty awful. Your highest speeds won't be effected much - those are all about air resistance. Sitting up and wide on a hybrid means that you lose more power to air resistance than if you were a bit lower and a bit narrower. You can actually get some of that with different setups, but I find it hard to make a flat bar bike comfortable like that, and my elbows still stick out; YMMV.
Anyway, I wouldn't worry about trying to go a lot narrower. 32 mm isn't too crazy a width. But if you have 20 tpi tires with extra rubber to prevent flats (Schwalbe makes some like this, can't imagine how they'd ride.

) you might be happier with 60 or 120 tpi tires with a lighter, more flexible puncture resistant device.
As far as what can go on your rim - Sheldon recommends a tire with an ETRTO width of 1.45 times the internal width of the rim as a minimum. I think that's a pretty good parameter - one wants a clincher to swell to a bit wider than where the bead hooks are to get a nice, firm connection, and it provides some protection for the rim. Hybrids are all over the place on rim width, so I won't speculate about what yours has. But it's likely to be on a sticker, or maybe stamped, on the rim somewhere. Or you can just take your tire off and measure.
IME, there is no advantage to higher pressure per se. For any given tire size, there are two good pressures for me - an on-road and an off-road pressure - and for a narrower tire, that pressure is higher. But the higher pressure is more of something I accept because I want a narrower tire.