At the time the Commission had 4 steamers fitted out as hospital ships and all 4 carried sick and wounded to St. Louis. See:The Democrat, St. Louis, MO, Tuesday, 20 May 1862, p. 2, col. 3.
As for when they brought the sick and wounded to St. Louis, see:
City of Memphis: Missouri Republican, St. Louis, MO, Friday, May 16, 1862, p. 2, col. 4
Empress: Missouri Republican, St. Louis, MO, Sunday, May 18, 1862, p. 3, col. 5, 6
Imperial: Missouri Republican, St. Louis, MO, Sunday, May 18, 1862, p. 2, col. 1
City of Louisiana: MissourDi Republican, St. Louis, MO, Tuesday, June 3, 1862, p. 2, col. 1
The D.A. January made the following 6 trips:
1 April 11, 1862 Pittsburg Landing April 14, 1862 Saint Louis 431
2 April 18, 1862 Pittsburg Landing April 23, 1862 Keokuk 284
3 May 2, 1862 Pittsburg Landing May 6, 1862 Cincinnati 300
4 May 10, 1862 Pittsburg Landing May 14, 1862 Jefferson Barracks 284
5 June 9, 1862 Pittsburg Landing June 12, 1862 Jefferson Barracks 375
6 June 19, 1862 Pittsburg Landing June 20, 1862 Paducah 459.
I haven't looked up the newspaper articles (if they exist) for these 6 trips.
Notice, the "City of Memphis" was not one of the Commission's boats. According to the first article the Commission was anticipating the "Red Rover" to be added to the fleet.