How to clean up ringing / noise on rising edge of CMOS output?
Your problem is that 74AHC1G04 is not a Schmitt-trigger inverter. It's an inverter without the Schmitt trigger input. You can see this in your second screen shot because the switching threshold on the downgoing edge is the same as on the upgoing edge.

The problem with not using a Schmitt trigger is that with the slow edges of your input signal, a little noise added to that signal can cause the inverter to see the voltage pass through its threshold level, then back across the threshold the other way due to noise, and then back again to the final level. With a Schmitt trigger the noise has to be much larger, on the order of 300 mV, before it can cause multiple transitions.

For a Schmitt trigger in the same family try 74AHC1G14.