Given $L$ is regular, the proof that $\mathrm{HALF}(L)$ is regular is pretty straightforward to me (e.g., #11 in this link): simply making a NFA and meeting in the middle with 2 original DFAs, the creation of such NFA proving it is regular.
However, what if you wanted to prove the 2nd half string is regular, not the 1st half of a string. That is,
$$\mathrm{2HALF}(L) = \{ x\mid yx\in L \text{ for some }y\text{ with } |y|=|x|\}\,.$$
Compare this to the standard
$$\mathrm{HALF}(L) = \{ x\mid xy\in L \text{ for some }y\text{ with } |y|=|x|\}\,.$$
I struggled on a solution, trying to use the same method as used on $\mathrm{HALF}(L)$ but can't seem to wrap my head around it.