In your specific scenario where a single process (or thread of execution) uses cooperative multitasking, you can use something like Windows' fibers or POSIX setcontext family of functions. I will use the term fiber here.
Basically when one fiber is finished executing a chunk of work and wants to voluntarily allow other fibers to run (hence the "cooperative" term), it either manually switches to the other fiber's context or more typically it performs some kind of yield() or scheduler() call that jumps into the scheduler's context, then the scheduler finds a new fiber to run and switches to that fiber's context.
What do we mean by context here? Basically the stack and registers. There is nothing magic about the stack, it's just a block of memory the stack pointer happens to point to. There is also nothing magic about the program counter, it just points to the next instruction to execute. Switching contexts simply saves the current registers somewhere, changes the stack pointer to a different chunk of memory, updates the program counter to a different stream of instructions, copies that context's saved registers into the CPU, then does a jump. Bam, you're now executing different instructions with a different stack. Often the context switch code is written in assembly that is invoked in a way that doesn't modify the current stack or it backs out the changes, in either case it leaves no traces on the stack or in registers so when code resumes execution it has no idea anything happened. (Again, the theme: we assume that method calls fiddle with registers, push arguments to the stack, move the stack pointer, etc but that is just the C calling convention. Nothing requires you to maintain a stack at all or to have any particular method call leave any traces of itself on the stack).
Since each stack is separate, you don't have some continuous chain of seemingly random method calls eventually overflowing the stack (which might be the result if you naively tried to implement this scheme using standard C methods that continuously called each other). You could implement this manually with a state machine where each fiber kept a state machine of where it was in its work, periodically returning to the calling dispatcher's method, but why bother when actual fiber/co-routine support is widely available?
Also remember that cooperative multitasking is orthogonal to processes, protected memory, address spaces, etc. Witness Mac OS 9 or Windows 3.x. They supported the idea of separate processes. But when you yielded, the context was changed to the OS context, allowing the OS scheduler to run, which then potentially selected another process to switch to. In theory you could have a full protected virtual memory OS that still used cooperative multitasking. In those systems, if a errant process never yielded, the OS scheduler never ran, so all other processes in the system were frozen. **
The next natural question is what makes something pre-emptive... The answer is that the OS schedules an interrupt timer with the CPU to stop the currently executing task and switch back to the OS scheduler's context regardless of whether the current task cares to release the CPU or not, thus "pre-empting" it.
If the OS uses CPU privilege levels, the (kernel configured) timer is not cancelable by lower level (user mode) code, though in theory if the OS didn't use such protections an errant task could mask off or cancel the interrupt timer and hijack the CPU. There are some other scenarios like IO calls where the scheduler can be invoked outside the timer, and the scheduler may decide no other process has higher priority and return control to the same process without a switch... And in reality most OSes don't do a real context switch here because that's expensive, the scheduler code runs inside the context of whatever process was executing, so it has to be very careful not to step on the stack, to save register states, etc.
** You might ask why not just fire a timer if yield isn't called within a certain period of time. The answer lies in multi-threaded synchronization. In a cooperative system, you don't have to bother taking locks, worry about re-entrance, etc because you only yield when things are in a known good state. If this mythical timer fires, you have now potentially corrupted the state of the program that was interrupted. If programs have to be written to handle this, congrats... You now have a half-assed pre-emptive multitasking system. Might as well just do it right! And if you are changing things anyway, may as well add threads, protected memory, etc. That's pretty much the history of the major OSes right there.