Get a smart trainer and start doing climbing indoors on the bike. Use Zwift or Rouvy or The Sufferfest or whichever training software you'd like, they all come with free trials. After a winter of doing this when you get outdoors in the spring you'll feel like a beast.
Everything else you are doing is also good, but I'll go out on a limb and say once you have a smart trainer with software you'll stop using the treadmill.
The best way to train for hills is to ride hills.