Cool project. I'm with Kim and bet you were really happy to see the end of those buckets!
In case a picture is helpful, here's an example of shoring jacks used to brace the sides:
https://www.mrblastit.com.au/wp-content/uploads/2015/02/Braced-Pool.jpg
To brace the ends, you could lay timbers (2x4s) across the top of the shoring jacks from end to end, all attached to one another and then attached to the shoring jacks (a couple of large pipe clamps at each junction) so the wood can't flex much. Two of these running the length of the pool would make sense. Everything will sag a bit, and then you can put straight props under each junction to hold everything up, level and also tighten it all up a bit. Imagine a grid of straight braces. That would be my choice, but I'm sure you'll get plenty of opinions
I don't know what's under there, but if you can drive a stake in the ground at the bottom around 8' out from the wall, you could brace with angled braces (three 2x4s per brace, with one angled down to the stake, one flat on the bottom, and one vertical against the wall. You could have as many of these as make sense, but probably two for each end wall. Similar spacing along the sides would work as well. If you have them meet in the centre, it's stronger because the angled braces aren't relying on the stake so much. (looks like upside-down rafters). I think it's harder that way because of the challenge of tightening it all up, but probably do-able with some wedges.
The flat bits against the wall, as already mentioned, spread the load, and should be around 2/3 of the way up from the bottom.
Just as an option not yet mentioned, and only if the bottom can be dug, you can make the pool deeper and order your new liner to fit the new shaped hopper.