You need TQFP burn-in sockets.
The OP has the chip soldered in. How exactly is a test socket going to help them with the problem? Those sockets are meant for programming/testing the chips
before they are soldered into boards, not after.
Those micro clamps work (I have also a set, in case I will need to do this) but they can be a pain to set up and it is easy to short two adjacent pins with them.
The easiest way is to find a nearest via or a passive component pad on the track in question and solder your wire there - those are usually a lot larger and easier to access than individual pins of the TQFP. In the worst case you can (very carefully!) scrape off a bit of soldermask on the track and solder directly to it too.
Of course, there are also specialized probing solutions like this:
https://sensepeek.com/(There was also a nice open source kit made out of aluminium extrusions with various types of holders and attachments but can't find it now).
However, if this is for a one off thing and you probe like this once in a blue moon, then you likely don't have the probe holder and buying/making one would be an overkill.