On a crop camera, I recommend the EF-S 55-250mm if on a budget - best value for the money. The 70-300 non-L is not really an upgrade in terms of IQ in my opinion and not worth the extra money over the EF-S. Even if IQ is ok when stopped down - I rarely had use cases for shooting at 200mm or longer at f8 or f11. It probably depends on what you shoot, but for me that rarely happens. AF is not much better, if at all, than the EF-S lens.
So for a real upgrade in terms of IQ and AF, it is one of the 70-200s or the 70-300L (cannot speak to the Tamron)