Examining company muster rolls for the period, as previously stated Company I showed from Oct. 17 to Dec. 31, 1861, station of company at Camp Wilcox. Several other companies showed the same, while Company H showed from Nov. 7, 1861, to Jan. 1, 1862, station of company at Camp Memminger which was near Halls Mill outside of Mobile. Someone else here with a detailed knowledge of the 23rd might have further information on the move of the regiment, or portions thereof, from Montgomery to Mobile.
Below is a link to a letter written by Martin L Calk of Company K, dated Dec. 9, 1861, from Mobile. Company K's muster roll, however, shows station of company for the period ending Dec. 31, to be Camp Wilcox. It thus appears that one or more companies, and perhaps the entire regiment had moved to the Mobile area prior to the death of Private Taylor. You will note that the letter refers to a measles outbreak, which could very well be the cause of death of Taylor.

http://digital.lib.uh.edu/collection/p15195coll31/item/1208