The 24-year-old joined Pools on loan from Blackburn Rovers on an initial one month deal in February before having it extended to May 15.
White made 10 National League appearances for Pools with his final outing coming off the bench late on in a 2-1 win over Dagenham & Redbridge on April 2.
Since then the midfielder – who is a free agent this summer – has had to watch Pools' promotion charge from the sidelines either as an unused substitute or out of the matchday squad entirely.
Despite White’s spell at Pools being slightly frustrating in terms of first team chances, Challinor still hasn’t ruled out a permanent deal in the summer.
"Tom is a victim of the competition we’ve got in midfield,” Challinor told The Mail.
“We’ll see where we are with Whitey because it’s the end of this chapter but it’s by no means the end if you like. We’ll see where things are for next season because he’s obviously being released by Blackburn.
"He’s been great here and it’s been difficult for him but he’s acted unbelievably well and professional in what he’s done. The opportunities he has got, he’s been effective in what he’s done.”
Explaining White’s notable omission from the side, the Pools boss added: “From our perspective, we’ve had an eye on the Aldershot game knowing he’d be going back after it and we wanted to get some continuity in the side going into the play-off games. He’s just a victim of that.”