That is a very old design cooler.
In the USA the Cooler Master 212+ or 212 EVO are the most common budget choice. They do use a backplate, which may require removing motherboard from case. EVO has a better quieter fan, if it matters. Height is ~165mm as I recall.
The Arctic 30i seems to get the best Brit/Euro price performance mention.
The CM TX3 used push pins mount, I think the latest version still does. This is a 90mm fan cooler. Both the Xigmatek Loki and the Zalman CNPS5X are similar and slightly better performers. Loki uses a backplate I think.
Thinking about this, unless you are folding or otherwise using CPU to 100% there should be no 24/7 problems.
There might be a case airflow problem keeping the stock cooler from performing properly. Or dust clogged case filters.
I would install a program that allows monitoring CPU usage and temperatures, though either a 212 or 30i cooler should be more than enough "cure."
Adding or changing case fans might be indicated.