These dates were theorized by Branch for June 25, 1859 to July 28, 1859,  not corroborated by Marleau 

June 25 Saturday – Sam piloted the J.C. Swon, (678 tons) under Captain Isaac H. Jones. Left for New Orleans.

July 1 Friday – J.C. Swon arrived in New Orleans

July 3 Sunday – J.C. Swon left for St. Louis.

July 6 Wednesday – Sam wrote to John T. “Tom” Moore from Memphis. Moore was a mud clerk on the Roe when Sam was a cub pilot there. The letter appeared in the Arkansaw Traveler July 14, 1883; the original has not been found and its authenticity is in doubt, though many elements argue for it being Sam’s [MTL 1: 91-2, n2; MTB 156]. Note: this may be the same Tom Moore that presented Sam for Masonic membership.
My Dear John:—

July 9 Saturday – J.C. Swon arrived in St. Louis

July 13 Wednesday – J.C. Swon left for New Orleans.

July 19 Tuesday – J.C. Swon arrived in New Orleans.

July 21 Thursday – J.C. Swon left for St. Louis. A sketch, “Soleather Cultivates His Taste for Music,” appeared in the New Orleans Crescent, signed by “Soleather,” attributed by Branch to Samuel Clemens [Branch, “A Chronological”; MTL 1: 93n2]. (See July 11, 1857 entry for Sam’s comments about the Rufus J. Lackand, and a note on Marleau’s recent research disputing these dates)

July 28 Thursday – J.C. Swon arrived in St. Louis.