1. Each person's TOS is different. Thus for any ONE of us to comment on what works for us, may NOT work for you. Understand this before endeavoring to apply any modality of help.
2. My personal recommendation is to get to know your anatomy and what is being compressed and how as reported to you by your doctor. If he/she can explain in simplistic terms so that you understand, great. Get a book on anatomy that might show the structures so that you can see how they are interconnected/related.
3. If you know the anatomy and the relationship of each muscle, vein, artery in your particular unique situation you may help yourself remedy the situation without surgery. I say "MAY" no guarantees. And of course seek professional advise.
4. I found value in the Anatomy coloring book AND Wolf-Heiddeger "The Color Atlast of Human Anatomy" The latter has excellant photos/drawings of the Brachial Plexus and nearby arteries/veins as well as different angles all up and down the arm, shoulder, neck and head. Should give you some excellant perspective.
5. Once you think you understand the anatomy, go back and make sure. Know that sometimes, not all, pain in one spot may not be created in that spot...it may not be the spot where the damage/injury is. This is called "referred pain." For example, pain in the wrist MAY be caused by compression on the Brachial Plexus in the neck. Hence one possibility that people with CTS do not get relief from surgery in the forearm. (some do if the compression is there).
6. Talk with a good PT about your particular situation/anatomy. Pick their brain and expertise on how best to help YOU. With me, I have two things. Pec Minor syndrome and Costoclavicular syndrome. Both sides (right and left) with the left being the painful side and the side where there is more compression. Mine is both vascular and neurologic in that both the nerves and the arterteries and veins are compressed. I was referred to an excellant PT here and between her expertise and my getting familiar with my body, I believe we BOTH have come up with MY unique treatment. Basically, MINE involves opening up my chest, stretches for the pec minor, strengthening upper back (mid traps and rhomboids) to HOLD my shoulders back so that the chest remains as open as possible. That is the basic theory behind all that I do. It has helped ME. Not saying it would help you, but many TOS have been helped by similar or related therapies since many are POSTURE related. i.e. weak rhomboids/upper back and tight chest, rounded shoulders, kyphotic upper back etc., etc. If this applies to you, ask the doctor/PT if these same types of exercises could IMPROVE YOUR condition.
7. YOU are your best professional help. YOU live in your body. The doctor/PT/Chiro etc., can only base their suggestions on what they see and what you tell them. They have experience and knowledge but they do not know everything. They may miss something because there is a limit to all that they can retain...sometimes you mentioning a particular thing may trigger their memory/knowledge of "oh, okay then you need to do X" You can help them to help you.

Good luck