Hi Suzie, welcome to the forum. We`ll be happy to assist you in your research.
Transfers between units were common during or upon completion of basic training and after the war ended but not when a man was assigned to his permanant unit during the war.
From the info you give, there are a lot of possibilities. Do you know the unit he took basic with?
Many men took basic in "provisional traing units" not assigned to any permanant unit. After basic they were assigned to new units being formed or recruited because of the skills for special units.
"Co. G 85th Mt Infantry/APO 345 c/o PM New York City (May 1945)"
APO 345 was for the 10th Mountain Division
The 85th Infantry Regiment was one of three regiments of the 10th Mountain Div.
The date of May 1945 is right at the end of the war and it suggests that this was his unit during the war.
HQ Co -- 157 Bn. 350th Inf APO 88 -- c/o PM New York City (Aug 1945)
APO 88 was for the 88th Infantry Division
The 350th Infantry Regiment was one of the three regiments of the 88th Inf Div.
the 157th Bn is a mystery. Battalions of the regiment were only numbered 1st, 2nd & 3rd. This suggests that the 157th is a separate non divisional unit attached to the 350th Inf Reg at that time.
The date of August 1945 is after the war ended & during occupation of Germany.
We dont know exactly were he was and when his transfers occured, but we have a good start on the possibilities.