When you grease you just want a small amount to pop out. Normally you can hear that. I used to find 3 pumps was good enough.
I placed a paper towel (other use old socks) between the stub axle & the disc to avoid contamination and then sprayed brake cleaner on the disc after greasing.
Bottom gaiters to cover the spring can keep the mess at bay but only fit when the area is clean.
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