Check what Anon-Co has to offer:
https://www.anon-co.com/
For instance the Tecsun radios they sell are very good. The PL-310ET would be good for backpacking as it's smaller than a paperback book and has a padded case. It only goes up to around 21MHz though, and doesn't have a BFO for sideband. On the plus side it's only around $40 with free shipping worldwide. The PL-600 does go up to 30MHz (29.9999) and has SSB. It's $57 plus $16 shipping to the US, so it would be approaching your budget limitation but slightly under. It's rather larger than the 310 also. They have a PL-380 that's $3 more than the 310, is the same size, has the same frequency coverage, and lacks an external antenna jack (at least the description doesn't mention one.) It does come with an external (probably clip-on) antenna and headphones though where the 310 doesn't include those.
A note about Tecsun radios: sometimes you get a lemon. Not often, but sometimes. If so, contact Anna at Anon-Co and she'll be happy to exchange it for another one (and she'll probably also have a wee chat with TecSun's factory about quality control as she's one of their largest dealers and has some clout with them.) And you can also ask her any questions you have about the radio you buy and if she can't answer it herself she'll contact Tecsun's engineers to get an answer for you.
It depends on what you plan to listen to: casual SW broadcast listening, in which case the 310 would probably do except for its upper frequency limitation, or also utility, military, ships at sea, hams and so on, where the 600 would be better.
I think I'd avoid their radios with analog tuning, no PLL and no DSP even though they're cheaper. You get what you pay for.
(FYI: the external antenna jacks take a 1/8 inch/3.5mm monaural plug, not a stereo plug that has tip-ring-sleeve. Tip connects to the antenna and ring/sleeve/whatever (the other connector) to ground or just leave it unconnected. That was the question I asked Anna which she had to refer to Tecsun's engineers.)