Hardware task switching was introduced on the 80286. And no modern OS - including Linux, except for the earliest versions - uses it, because the way it is implemented, the mechanism will do a lot of unnecessary duplicate work (preserving "garbage" register contents from just before the kernel is about to return to user mode, and will consequently have to restore them itself).

The only situation where it would be required to use this misfeature is for handling exceptions like stack overflow in the case that they can occur in kernel mode. A "Task State Segment" is really more like a pre-allocated stack frame anyway, the CPU puts active ones in a linked list and does not allow freely switching between them, only returning to the outer level.

x86-64 does not have hardware task switching anymore, instead it is now possible to switch to another stack on an exception or interrupt, which is all that the TSS was used for in practice anyway.