Congratulations Mike on your weight loss, that is something to celebrate, lifestyle change isn't easy and takes time. Mike, I'm by no means a personal trainer, or health guru so take what I say as my opinion, but I can tell you as a son of a lifelong endurance and fitness proponent and trainer that body weight exercises and CONSISTENCY in whatever you chose to do is going to be most beneficial to you right now, both in your nutrition and your exercise. If you've been sedentary for an extended period of time simply getting out like you've been doing and walking paired with your change in nutrition is going to yield an increase in overall health. I'm going to be the devils advocate here and disagree with the once-a-week approach to preparation for an extended hike, sure, there is no argument to the benefits of doing so and it works for some, but you run the risk of overuse injuries biting off more than you can chew in an extended day hike at this point. Gradual growth and progression throughout the week is going to train your body's muscles, bone structure, ligaments, tendons, etc. for the relative rigors of repeated use on the trail hiking which you'll encounter in your GA section hike. Once you've established a solid base level of fitness I think you'll be ready to start training with more specificity towards your goal, but in your early phases (which from your post it seems you are in) progressive and consistent proper physical activity throughout the week is going to be most beneficial.
It's been my experience that many traditional gym trainers have one goal in mind and that is to LOSE WEIGHT, which seems harmless enough and is a byproduct of attaining fitness, which should always be the foremost goal IMO. Often trainers neglect functional strength, which can very easily be attained in simple body weight exercises that can be done at home or on the trail such as: hiking, sumo squats, lunges, pushups, pullups, chinups, short intense cardio intervals, box jumps, planks, burpees, knee raises. I'm not suggesting you quit your trainer, just consider that being able to move your body around with control and function isn't always found in repetitions on a machine that limits your range of motion and controls the weight for you. Eric nailed it in his suggestion to just get out and walk hills, nothing can really replicate hiking like...hiking, pair that with some cross training and you'll lose that weight you mentioned, be ready for your section hike, and have some additional fitness to boot. Elevating your heartrate is going to be beneficial in gauging your effort, just recognize your limitations and don't push yourself to the point of injury. You're probably going to get a "forget all this workout mess and just hike!" suggestions, sure, hike and hike as often as you can, but just hiking doesn't necessarily develop a well rounded and fit individual, but that's an entirely different subject I suppose. Best of luck to you Mike!