As John Dallman points out, it is common for a Secretary to continue serving in their post despite a change in President. It is far less common, however, for a Secretary to reprise a role under another, future, President - as would be the case if Vilsack is confirmed.
There are a few, but it's pretty rare. There are only six reprisals of Secretary posts that I know of, the only one in living memory being Donald Rumsfeld's nomination to the post of Secretary of Defense.
Secretary of State
- Daniel Webster - Nominated by Harrison in 1841, subsequently by Fillmore in 1850.
- William Hunter - Nominated by Pierce in 1853, subsequently by Buchanan in 1860.
- James G. Blaine - Nominated by Garfield in 1881, subsequently by Harrison in 1889.
Secretary of the Treasury
- Hugh McCulloch - Nominated by Lincoln in 1865, subsequently by Arthur in 1884.
- William Windom - Nominated by Garfield in 1881, subsequently by Harrison in 1889.
Secretary of Defense
- Donald Rumsfeld - Nominated by Ford in 1975, subsequently by Bush in 2001.
If we count Attorney General as a Secretary-level position, there have been another couple:
- John J. Crittenden - Nominated by Harrison in 1841, subsequently by Fillmore in 1850.
- Bill Barr - Nominated by Bush sr. in 1991, subsequently by Trump in 2019.
Finally, if we include posts that are no longer extant, or are no longer Cabinet members, there are another couple of examples:
- John Mason - Nominated for the position of Secretary of the Navy in 1844 by Tyler, and subsequently by Polk in 1846.
- Henry Stimson - Nominated for the position of Secretary of War in 1911 by Taft, and subsequently by FDR in 1940.