Do you have a make/model for the pc?
Or, perhaps a photo of the space you have to work with?
Does your current cooler now have a fan attached, or is it just a heat sink?
Any cooler needs a source of fresh air to let it do it's job.
If there is no fan to help get air into where the cpu and cooler are, nothing will really help. You could limit the max performance in windows from 100% to perhaps 80% to get the heat down.
a 70mm fan needs to run at very high rpm to do any good.
There are some 80mm noctua fans tat are relatively quiet.
Here is a link to some low profile coolers.
https://www.frostytech.com/top5_lowprofile_heatsinks.html
You should note that the stock intel cooler is quite decent.
There are several types with differing heights.
The ones with copper bottoms and are taller are the strongest.