The observed performance difference on the power supply was actually caused by the difference in rising edge detect (flashlight turning off, tailcap turning on) versus falling edge detection (tailcap on, battery running down). The spec’ed cutoffs for rising and falling can be seen at the end of this translated datasheet.
Unfortunately a capacitor cannot change the system voltage, it will only mimic the battery voltage. And there’s already a capacitor in place.
In short, the HM1160 is unfortunately a “take it or leave it” scenario. It’s handy, but not flexible. For flexibility, you’d need something like one of my smart tailcaps which is running an MCU and can be programmed with any voltage cutoffs as desired. Or… I’m currently finalizing a tailcap that is similar to LoneOcean’s tailcap (in concept and functionality, not design). It’ll use 4 red/blue bi-color LEDs and a couple low-power voltage reset monitors to switch from blue to red to off at whichever cutoffs are desired (in 0.1v increments). I hope to have it ready in the next few weeks.