For your reference, the change should be made here:
if(st < el_height/2)
by modifying the /2
as well as here:
newH = el_height/2;

for my purposes, I changed to /3.

Hope this helps others.

Guessing somewhere in this piece of js?:

   function avia_header_size()
    {
        var win				= $(window),
            header          = $('.html_header_top.html_header_sticky #header'),
            unsticktop		= $('.av_header_unstick_top');
            
        if(!header.length && !unsticktop.length) return;
        
        var logo            = $('#header_main .container .logo img, #header_main .container .logo a'),
            elements        = $('#header_main .container:not(#header_main_alternate>.container), #header_main .main_menu ul:first-child > li > a:not(.avia_mega_div a, #header_main_alternate a), #header_main #menu-item-shop .cart_dropdown_link'),
            el_height       = $(elements).filter(':first').height(),
            isMobile        = $.avia_utilities.isMobile,
            scroll_top		= $('#scroll-top-link'),
            transparent 	= header.is('.av_header_transparency'),
            shrinking		= header.is('.av_header_shrinking'),
            topbar_height	= header.find('#header_meta').outerHeight(),
            set_height      = function()
            {	
                var st = win.scrollTop(), newH = 0, st_real = st;
				
				if(unsticktop) st -= topbar_height; 
				if(st < 0) st = 0;
				
				if(shrinking && !isMobile)
                {
	                if(st < el_height/2)
	                {
	                    newH = el_height - st;
	                    if(st <= 0){
							newH = el_height;
					    }
	                    
	                    av_change_class(header, 'remove', 'header-scrolled');
	                    //header.removeClass('header-scrolled');
	                }
	                else
	                {
	                    newH = el_height/2;
	                    //header.addClass('header-scrolled');
	                    av_change_class(header, 'add', 'header-scrolled');
	                }
	                
	                if(st - 30 < el_height)
	                {
	                    av_change_class(header, 'remove', 'header-scrolled-full');
	                }
	                else
	                {
	                    av_change_class(header, 'add', 'header-scrolled-full');
	                }
	                
	                
	                elements.css({'height': newH + 'px', 'lineHeight': newH + 'px'});
                	logo.css({'maxHeight': newH + 'px'});
                }

This semi-worked, but it still minimizes all the way (43px max height?) and then bounces up to the 60px max height.