I resumed riding about a decade ago, after 20 years out of the saddle. I started with weekly lessons, and then moved up to twice weekly. I was making progress, but slowly. One I felt comfortable riding unsupervised, I went to a half lease (3 days a week at a second barn) in addition to my 2 lessons a week at the original barn. Then I went to full lease at the second barn, eventually dropped the lessons at the original barn, and now have a horse full-time under my care.
IME one lesson a week is not going to really have you progress very fast, or even much at all. Your barn half lease sounds like a very good deal, three rides a week including a lesson. If you feel safe to tack up and ride without supervision, I would take that lease.
You can also work on your overall fitness at the gym, which may help a bit.
I had no lessons as a kid. Coming back as an adult via lessons, I learned how dependent you can get on the lesson format and supervision. I think it’s important to start riding on your own as soon as possible, so that you can start thinking through what you are doing, what your horse is doing, and listening to your horse as well, rather than letting the instructor take precedence over your own feelings and intuition.
As far as hacking out, that probably depends on the physical location and culture of the barn you are at. Around here, many lesson barns are “landlocked,” in the middle of a district of working farms, with no trail access, unless they happen to be close to a park or greenway. And many lesson barns or show barns don’t see hacking out as necessary or safe.
For me, it’s absolutely essential, so a big reason I never ended up with a horse at the orignal barn I took lessons at, was that it had no trails access.