Easiest free speed is to get more aero. Lower position. Snug fitting, less flappy clothing.
Then tires. The size -- 700x32 -- doesn't matter much if the tires are sluggish. I recently switched my hybrid from Continental Sport Contact 2 700x32 back to Conti Speed Rides (700x42), and picked up another 1-2 mph just from the tire switch. The specs seem similar -- similar weight, etc -- but the Sport Contact 2 tires felt harsh at full pressure and sluggish at a comfortable pressure. The Speed Rides are better tires overall, faster, better grip, more comfortable and more versatile. I use heavy, puncture resistant tires on my errand bike/comfort hybrid, but I don't want them on my fun bikes.
That's good for a little more speed without major modifications or switching to a road bike.
I also switched my hybrid over the past few years from the original flat bars to riser and now albatross swept bars. The albatross bars are comfortable and more versatile, with upright position on the grips and more aero in the forward curved part of the bar. That's my casual and group ride bike. Over the usual 20-50 mile rides I average 12-14 mph.
Better physical conditioning can help a bit. But the faster we go, the more wind resistance becomes a greater factor. Beyond around 14 mph it takes a lot more energy just to overcome wind resistance, whether lack of aero positioning or headwinds. Makes more difference than bike weight, rotating wheel/tire weight or the type of tire. That's the advantage to a drop bar road bike. But I don't always want to ride my road bikes so my compromise was albatross bars on the hybrid.