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I bought an ankle brace the first time I sprained my ankle thinking I was just going to use it for that injury, but with the frequency of minor sprains in BJJ I've ended up busting it out every 6 months or so for the past 5 years. Probably saved money vs tape and I didn't have to learn how to tape the right way, which I think is kind of involved, or pay someone to do it.
Never tried it but they seem a bit low around the ankle. If I were you I'd pair it up with some sort of ankle stabilizer
Yes, I can confirm that the ASO is excellent - very supportive and comfortable to run in.
It's probably best if you go into a local shop and try on a variety of options. Taking online recommendations can be a crap shoot.
That said - I like trail runners for most all conditions. They are cheaper than traditional leather boots and more comfortable. Salomon, Brooks, Altra, and New Balance make some good options.
Asolo 520's are my favorite traditional boot. Check em out.
If you need serious ankle support, I would look at braces instead of using a boot. I use ones like this inside my trail runners when necessary:
I'd personally recommend the ASO cast. It fits into any shoe and is very adjustable in terms of support.