I think the 11-34 cassette would be the minimum worthwhile change. You would definitely notice the difference. For reference, second gear on the 11-34 cassette is 30T, so it basically gives you 1 full step lower gear.

One of my bikes has Shimano 105 with a compact 50/34 front and 11-34 cassette and I use the 34T on climbs steeper than 10% or longer alpine climbs above 6-7%. It's not a massive difference, but signficant. But if you want to be truly spinning up steep climbs like you might on your mtb, then you need to look at gravel bike gearing with a 40T+ cassette and smaller chainrings e.g. Shimano GRX. But that's a whole new drivetrain. I think the biggest cassette you can fit on current 105 without modification is 36T (SRAM 11-36T is compatible).