I don't know right off hand of anything which would indicate one way or the other as to the disposition of the men of Hart's Battery.
If I were to make a guess I would say that the majority of Hart's command remain east of the Mississippi probably with their original units the 24th Arkansas.
AS you are aware the 19th and 24th Arkansas were supposed to be exchanged and reunited with those not captured at Arkansas Post. However instead them being exchanged at Vicksburg, as was normal for southern Trans-Miss Prisoners, Gen'l Grant did not want those exchanges reenforcing the Vicksburg Garrison is the reason that they were exchanged at City Point. That is how Hardy's 19th/24th Arkansas west of the Mississippi came into existance.
Because of the manner of the Surrender of Arkansas Post, no one wanted these men. Even though Robert E. Lee was screaming for more Western Troops in Virginia he turned down these troops. Only Pat Cleburne would accept them. Because large numbers of troops by May 1863 could not cross the Mississippi to return to Arkansas because of the Union Navy, these men were generally stuck in the A.o.T. with Cleburne.
While it was possible for a small group to evade the Picket boats and convoys on the Mississippi, like William Hart returning to Arkansas, I doubt that his own command would have been able to do so.
Just my thoughts on the matter.