I would go 1×10 and replace your shifter, derailleur, cassette, chain ring and chain. I did this on a 2×9 bike of mine last year and it cost me $180 all-in (SRAM GX shifter & derailleur). I am sure you can get it cheaper now. The NX range is even lower cost. I figure for $150 you should be able to get it done. More than worth the money.
The 11-40 9-speed cassette in one of the links above alone are $139 before taxes/shipping. You can get a 10-speed Sunrace cassette with that spread for $65. And you can go as low as 30t on the chainring. A 30t chain ring with 40t low gear on the cassette gives you a pretty low ratio and you should be fine with that almost anywhere.