This cap belonged to an officer and I would have expected Insignia on the front of the cap as well but I could see no evidence of such. Perhaps the front insignia had be removed. In photos of the 1st Michigan Three Month Infantry, taken in Detroit, some officers wore no Insignia. I assume Detroit did not have enough Insignia on hand and that the officers would have added Insignia once they got to Washington.
Being from Michigan I am on the look out for uniform items made in Michigan. This captain was original in the 1st Michigan Three Month Infantry so his first cap would have been made in Detroit. I am guessing this cap is not the one made in Detroit. His vest is in the glass case as well. It is possible his vest was made in Detroit. His jacket is in another museum and I am not sure if it was made in Detroit.