For maybe $60, you can get a Deore-level 9s 12-36 cassette, which I have on my mountain bike. 9s chain is a bit more expensive than 8s, but very much cheaper than 10s. The 36t cog should work fine with any vaguely recent Shimano MTB derailleur, and the new ones are designed for it.

Since a 53x12 top gear is silly, you might swap the 12 for a 13 bottom to give you something like 13-14-16-18-21-24-28-32-36 cogs. This isn't much different than your current cassette -- there's just a moderately big cog and huge dinner plate added.

39x36 should get you up pretty much anything.