Both my hybrids have wider than spec tires on 622x19 and 622x17.5 rims. Seating beads was a bit tricky at first.

My main trick with stubborn tires is to inflate partway with the tires still soft and roll them on the ground while mashing downward. I'll roll both directions so the valve doesn't squiggle off kilter. Then pump some more, check again, etc. But this is more important with loose fitting tires so they don't blow off the rim or the tube doesn't sneak out between the tire and rim. I quit using any tires that didn't fit snugly enough to allow rolling the bike on a flat tire.

And FWIW I use a Topeak RaceRocket HP mini-pump, including on my hybrids. It's intended for high pressure skinny road bike tires but works fine on the hybrid tires. I can pump up to 40-50 psi in 200 strokes. Easy strokes, not wrist breakers. Just takes longer with the low volume pump. I might get the mountain bike version of the Topeak RaceRocket. Good little pump for being so lightweight and small.

I run 700x40 Michelin Protek Cross Max on 622x19 rims, which are spec'd for 700x38 max, and the Michelins run wide, closer to 700x45. They work fine but seating the bead is fussy. The tires are massively thick, including the sidewalls and beads, but they're bulletproof so it's worthwhile for my errand bike.

First, I have to use heavy duty rim tape to keep the tubes from extruding through the deep nipple holes -- thin rubber rim strips won't support the tube well enough. I already had problems last year with tubes extruding and eventually developing tiny splits and slow leaks. So I'm using cloth tape (Velox on one, Nashbar on the other -- cloth rim tape is all the same).

The tricky bit is the width. The first roll of cloth tape, I had to cut lengthwise in half, which was a hassle but I needed it done ASAP. I bought tape measuring the full width of the rim channel, based on watching some YouTube videos. Big mistake. This prevented the wider than spec tires from seating. So I cut the tape lengthwise to fit the narrow recessed channel. And I bought narrower cloth rim tape for the next wheel. No problems after that.

Well, mostly no problems. Whenever I need to remove and redo the tires for whatever reason, the Michelins don't seat quite perfectly until I pump 'em to maximum pressure, when the slight off-kilter edge pops into place. Sometimes riding around the block helps. Then I reduce the tire to my preferred riding pressure.

If I had to do it again, I might use hard plastic rim tape rather than cloth to support the tubes with double wall rims.

The other bike has Araya PX-45 rims, which measure 17.5 bead to bead inside. I think they were originally spec'd for up to 700x35. They'll handle 700x38, no problem, and I'm actually running 700x42 (nominal only) Continental Speed Rides, which measure closer to 700x38. The Speed Rides have thinner, flexible sidewalls and a normal bead, and these are single wall rims so I've been able to get by with ordinary thin rubber rim tape.

Occasionally when I've fixed a flat and used my mini pump during a ride I'll pump only to 40-50 psi. At first one bit of the tire will not seat quite perfectly with a little hitch visible in the reflective strip. But as soon as I ride it a short distance it pops into place.