What can reduce overshoot and ringing on a simple square wave pulse generator?
From the looks of the new scope traces added to the question, specifically the Vcc trace, it appears that the ringing is originating in poor regulation of the supply at the point of use - most likely not at the bench supply output. While shorter leads from the bench power supply will certainly help by reducing lead inductance, that won't be enough when the transition is as sharp as you're looking for.


Add a hefty capacitor on the breadboard across the supply rails, closest to the IC: Start with 100 uF. 
In parallel with the 0.1 uF decoupling capacitor shown in your schematic, and touching the Schmitt Trigger supply pins, add a 10 uF electrolytic capacitor.
Trim the leads of all 3 capacitors above to the bare minimum which will still make positive contact with the breadboard contacts. Those leads are adding inductance you don't want.
Add a load from the output you are reading to the ground pin, as close to the output pin as possible - 220 Ohms should be fine, and again you want leads trimmed to minimum.
If you absolutely must avoid overshoot / undershoot beyond a few hundred milliVolts, add small signal Schottky diodes from the output pin to both supply and ground pins, thus:







simulate this circuit &ndash; Schematic created using CircuitLab  


This will ensure that the peak on the rising edge and trough on the falling edge of the ringing are damped - there will be some effect on the respective trough / peak of the ringing as well because of the excess energy of the peaks being dissipated across the diodes.
Finally, the breadboard, due to the nature of its construction, introduces capacitance, inductance, and all kinds of parasitic coupling. Even a simple perf-board will do better. Long leads simple exacerbate this problem, especially at high frequencies / sharp transitions, where even a simple wire lead is a source of coupling and inductive ringing.