I’m in a similar bing, and my thought process *might* help.
I was running 22/32 on 11-34 which did me for everything (not amazingly fit). But had the opportunity for an upgrade and ended up running 28/42 on 12-36… Quite a difference and I can’t spin how I want to on climbs.
3 options. Larger cassette, smaller chainring, or 1×10 with larger cassette.
Can’t do a smaller chainring without needing a new rear mech as the gear jumps will be too big.
Install a 40t cassette expander and go to a 30t chainring. This gives me a slightly lower ratio 1st gear than I had before (0.7 vs the previous 0.64). So would probably do the job. However, cost incurred is about £50 for a cog and £50 for a spiderless chainring (SRAM cranks)
Finally, the option I’m going for, at least initially. Keep the chainset 28/42. Add a 40t Hope T-Rex on the back which gives me the same 1st gear I had before the change, but still allows me the same at the other end. Total outlay about £50. (Plus new chain since the old one is too short).