MO SOS file has him in Co K, 6 Provisional Enrolled Militia, First Sargent.
Card is here: https://www.sos.mo.gov/Images/Archives/Military/s00819/s00819_2318.pdf
Enrolled in Rolla Mo July '63. C K, 6th PEMM was raised from Co L, 73rd MO EMM. It was a mounted unit.
Baskett's original enrollment is in Co H, 73rd EMM as a private on 25 Oct 1862 again at Rolla.
It appears that he, William and his son Charles M. Baskett were from Texas County Missouri, found in the 1860 census there with wife Mary (Turpin) Baskett. Post war family ends up in Washington State where both gentlemen receive pensions for their service in the 6th PEMM. William's grave entry is here: https://www.findagrave.com/memorial/5101137/william-baskett
Census records indicate he was born in 1820 in Tennessee. He is found in Shannon Co MO in 1850 census then in Texas Co in 1860, 1870, and 1880.
This fellow should not be confused with William H Baskett of Co A, 1st MO Cav, a confederate unit. He was from Arkansas and transferred to 1st Cav from 1st Ark Sharpshooters. Wounded and captured at Franklin TN.