Question

The idea is to move a div downwards slightly when hovered and when clicked to move further down to reveal content. The problem is that when you have clicked the div and it moves down you are no longer hovering on it so it performs the ‘close when not hovered function.

$(function () {
    $('.more').hover(function () { //Open on hover 
        $('#pull_down_content').animate({
            'top': '-360px'
        }, 1000);
    }, function () { //Close when not hovered
        $('#pull_down_content').animate({
            'top': '-380px'
        }, 1000);
    });
});
$('.more').click(function () { //Move down when clicked
    $('#pull_down_content').animate({
        'top': '0px'
    }, 1000);
});

Answer 1

Put a class on the element to signify that it has been clicked, and do not close it if it has that class:

$(function() {        
    $('.more').hover(function(){    //Open on hover 
           $('#pull_down_content').animate({'top':'-360px'},1000);
        },    
        function(){   //Close when not hovered
           if (!$('#pull_down_content').hasClass("expanded"))
               $('#pull_down_content').animate({'top':'-380px'},1000);    
        });
    });
    $('.more').click(function(){    //Move down when clicked
        $('#pull_down_content').addClass("expanded").animate({'top':'0px'},1000);
    });

Of course you now need another trigger to determine when exactly to undo the hover animation after the item has been clicked; how to do that exactly depends on the effect you want to achieve.