Yes tighter support can help. Is it just the left testicle or both? In about 85% cases it appears in only the left. If your dr has recommended surgery it is probably not going to be helped by wearing tighter support as it is too severe. But then again he may be over-reacting.
Dude, don't squeeze it. It's veins after all and you can do other damage such as inhibiting blood flow.
Besides surgery there is embolization which basically reduces pressure in the veins thus reduces size. Laparoscopy is a less invasive option to surgery but as with surgery it also has possible complications.
Varicocele can cause infertility so it probably best to get sorted out sooner rather than later.
Good luck.