Issue when adding a single pole switch next to a 3 way
REWRITE:
Having the 3-way travelers go through the fan box is quite unusual, but I was &quot;going on faith&quot; on that.  However, given the behavior of the supposed &quot;supply&quot; wires you mention in comments, notably white being hot and black being pulled down to 0V... it's clear this is a 3-way switch loop.  Neutral is not present at the box at all.
I recolored it to designate wire function (black=always-hot, red=switched-hot, yellow=travelers)... because 3-ways are too confusing otherwise.

Note the complete absence of neutral from this box.  White is an always-hot wire (as is required by Code; if neutral is not present in the cable and always-hot is; white must be used for it; that makes it easier to detect as a hot.  It's odd that the person knew this rule yet did not use it in the other switch leg).  Black in your circuit is switched-hot to the fan - i.e. it powers the fan.  The fan's load is &quot;pulling it down toward neutral&quot; which is why it reads 0V.  It is not neutral.
If you have any doubt of this, connect that black and white to a receptacle and plug in a 1500W heater or toaster (simple heat appliance, but high in wattage). I guarantee the fan/light will activate if you do.  Switch loop!
Neutral is not present in an old style switch loop like this, so branching an additional light off it is not going to happen. Sorry!