I found a great solution here:
https://mcmap.net/q/720521/-once-i-click-on-label-select-button-should-get-open
In your case you can wrap the button in a label and apply the same solution.
HTML
<label for="orderby">
<button type="button">
<span>Sort</span>
</button>
</label>
<select id="orderby" name="orderby" class="orderby" aria-label="Sort By">
<option value="date" selected="selected">Latest</option>
<option value="popularity">Popularity</option>
<option value="rating">Average rating</option>
<option value="price">Price: low to high</option>
<option value="price-desc">Price: high to low</option>
</select>
JS
$("label").mouseover(function(){
var $this = $(this);
var $input = $('#' + $(this).attr('for'));
if ($input.is("select") && !$('.lfClon').length) {
var $clon = $input.clone();
var getRules = function($ele){ return {
position: 'absolute',
left: $ele.offset().left,
top: $ele.offset().top,
width: $ele.outerWidth(),
height: $ele.outerHeight(),
opacity: 0,
margin: 0,
padding: 0
};};
var rules = getRules($this);
$clon.css(rules);
$clon.on("mousedown.lf", function(){
$clon.css({
marginLeft: $input.offset().left - rules.left,
marginTop: $input.offset().top - rules.top,
});
$clon.on('change blur', function(){
$input.val($clon.val()).show();
$clon.remove();
});
$clon.off('.lf');
});
$clon.on("mouseout.lf", function(){
$(this).remove();
});
$clon.prop({id:'',className:'lfClon'});
$clon.appendTo('body');
}
});
Demo:
http://jsfiddle.net/Yh3Jf/24/
The js creates a clone of the select element which updates the original.
Tested and works for me, in my case I used it to hide the select field with css, then the button triggers a drop down where a selection can be made.