Kea, it looks to me like the Post Office’s biggest box is 11" X 8.5" X 5.5". There is also a 11-7/8" x 3-3/8" x 13-5/8" size box but that is less than 4 inches thick.
Not sure how or when you would need to mail the extra equipment. You can buy other size boxes at department stores (Walmart etc.) and you can mail thoses boxes throught the US Post Office. You do not need to use the US Post Office boxes to mail your extra equipment.
I would think twice about sending spare tent poles etc. along the trail. I suggest you keep anything extra at home. (Not sure where that is.) In the case you do have some type of emergency you can purchase items along the trail. Even a broken tent pole can be replaced on the trail within 3-4 days. You would spend more money on postage sending items you may never use along the trail.
Clothing and sleeping bag would be the only big items that would need to be changed out if you are hiking early spring to early fall. Not sure where you are hiking or how long. You can send me an email and discuss this or continue posting on Trail Forums. I will be glad to help out as much as I can. When are you doing your hike?
bilko