One thing here, " lower/distant knocks" are different situations.
Lower can mean standing right over the piezo and tapping foot lightly.
Distance could mean standing in entrance of room slamming or tapping foot on floor, compared to standing 1 foot from device doing the same.
How to tell if it is just a lower knock? Or if it is a knock of distance?
For distance maybe 2 or more piezos a distance apart, compare the readings, difference of readings give distance from primary piezo pickup.
Do you have a link to the info of the circuit you posted?
Maybe you could condition the piezo signal like described here
, gives one nice pulse.