You can set the CSS property
pointer-events: none on the obstructing object... but be aware that pointer events are all or nothing; mouseovers and hovers will pass right through, but so will clicks.
Here is a description of the value, from the Mozilla Developer's Network:
none: The element is never the target of mouse events; however, mouse events may target its descendant elements if those descendants have pointer-events set to some other value. In these circumstances, mouse events will trigger event listeners on this parent element as appropriate on their way to/from the descendant during the event capture/bubble phases.
I've put together a little example. In this example, I'm using
onmouseout, since that's what you use on your website, but you could just as easily use the CSS
:hover pseudo-selector. Here's a jsfiddle version of the example, and the stack snippet is below.
/* this first line is the important part */
<div class="hoverable" onmouseover="this.style.backgroundColor = 'green'" onmouseout="this.style.backgroundColor = 'blue'"> </div>
<div class="obscuring"> </div>