I personally don't think hand winches are worth owning for most people. A good one costs as much as an electric winch and usually has less power and less range. For tasks other than vehicle recovery they're great. I have a couple for utility use.
The biggest downside of electric winches on a budget is that it's expensive to attach them. There are a number of options though. You already have a hitch - so a hitch mount winch is one good option. It's not like you're getting stuck a lot - so pulling out from the back isn't always ideal, but it probably would work most of the time for most situations. There are also some "winch in a box" models for around $300 that have straps that attach to your tow points. Like this
https://www.amazon.com/Superwinch-1...qid=1498231705&sr=8-2&keywords=Portable_Winch
The cheapest option would probably be a harbor freight 5k winch with a snatch block mounted on a receiver hitch and a set of 4ga jumper cables hard wired to the winch for power.
I'm surprised we don't see Dewalt or ?? 20v portable winches. The batteries are getting so good now that they could work very well for at least one pull. If the max load is 200 amps at 12v (which is rarely maxed out), a 20v 6ah battery would last for 3 minutes of pulling or roughly 30 feet on a typical 9500lb winch. But since it's almost never going to reach max load, you could probably get more like 1-2 full 80 foot pulls out of a battery. For 99% of us that would get us unstuck and on our way.