The conventional x86 way of doing it was to spill to "the stack" and I'm told that their chips actually specially optimize that nowadays. But it's all very weird, deep magic in a lot of ways.
The only stack-specific special optimization that's done is fusing the decrement/increment of esp/rsp with the store µop. And that's done mainly since push/pop are one byte opcodes, unlike general load/store.
Everything else is general memory optimizations that apply for everything like the aforementioned store forwarding. It's still expensive if the CPU can't use them (mismatched load/store size, incorrect speculation, etc.)
