First, congrats on taking your first PT. Now, ignore that score.
No, seriously. It's great to know where you're starting, and have a decent sense of the exam and what your natural strengths/weaknesses are, but please don't think that initial diagnostic limits you in any way. No one can predict for you whether you'll be ready by October, but it's 100% in the realm of possibility. You won't be able to make any sort of reasonable prediction until around 2-3 weeks out from the exam.
At that point you should assess what's reasonable to focus on in 2-3 weeks, and decide whether you want to postpone until the December. All that decision-making is for September, not now!
Remember that people have a huge range of experiences when studying for the LSAT. Some people will have linear improvement bit-by-bit over the long haul. Others will see zero improvement for a month or more, then a sudden leap of 5 points, followed by another frustrating plateau. I've watched a number of students move from the mid 150s to 170+ in four months, though it's certainly not the 'average' experience, and it requires a great deal of work.
Are you not able to start studying before June 1? If you're not, then fine, but if you are able to I'd kick off at the first opportunity. If you take a prep class, don't wait until the first session to get cracking.
It's very normal to feel exhausted after the break on your first PT. Building up your endurance is a real issue. Working out will help some, indirectly, but you will also need to directly build up
mental endurance, by
mentally working out and stretching yourself to and past your limit.
I generally view LSAT preparation as existing in three overlapping phases: comprehension, efficiency, and endurance. These aren't attacked individually and consecutively; rather, the prioritization of these three items shifts throughout the course of LSAT prep. The beginning of LSAT prep is focused almost entirely on building comprehension foundation - how can you possibly hope to execute a logic game perfectly in 8 minutes if you can't get to the right answers even with unlimited time?

As you build that foundation, you start thinking more and more about how to do the awesome breakdowns/diagrams/inferences/etc more and more efficiently - how to recognize common patterns faster, how to read in a way to zero in immediately on the conclusion, etc. Then, after you've got a solid framework for doing the right work (comprehension) and doing it efficiently, you'll start to focus on making sure you can do all that awesome stuff over the course of 5 sections: endurance!
You'll be building up your mental endurance just by studying, assuming you don't study in 15 minute bite-sized chunks the entire time. When you get closer to test day, and PTs become the centerpiece of your prep, you'll be actively building up precisely the endurance that you need. To make sure that you're stretching yourself, I often recommend that my students start doing 6-section exams near test day. It's a lot, but if you can get used to that, you're endurance should be great on test day.
All in all, you've got a nice chunk of time to do some serious damage. Make sure you pace yourself - find the balance between slacking off and burning out. You'll be a rockstar.
