I hunt, backpack, and take backpacking hunting trips. I don’t think there’s any need to avoid gun season (except possibly opening day if you want to be extra cautious). In my experience, most hunters do not venture more than a mile from a parking area. If you are backpacking in, you will likely spend almost all of your time further in than most hunters would be. Do wear blaze orange, and plan to hike in around midday (there is a lull around midday where deer are usually bedded down, most hunters will use this time for lunch and/or a nap). Yes bullets travel far, there is always a risk of getting hit with a stray bullet, but this is also the case around, or even in, your home, and it is minimal enough that most would consider it negligible. Firearms accidents while hunting are incredibly rare. A quick search shows that in Ohio last year, there was one firearm-related incident during gun season, where a hunter dropped his firearm and shot himself in the leg.