The z-index will only change stacking order, not the x & y positioning.
Relative positioning can be used to offset elements in x y space and create an overlap. But you may have to think carefully how you apply the offsets to keep responsive.
I would like to throw in “position:absolute” along with “position:relative”, or possibly negative margins. And repeat the caviat that you need to consider how elements are resized at different widths.
The “best” method to use depends on context… the rest of your page and how the positioned elements are used.