Not sure of your brewing method but to minimize waste/ ecological & economic burden, just save the graywater from any rinsing regardless of filter material. If I have to use a brewer with a cone/cup filter, the whole filter & grounds assembly goes to compost that day, but I prefer the flavor & ease of the aeropress; immersion compared to drip seems to get better extraction & thus saves money/ shipping fuel/ ecological burden on coffee, too.
An aeropress uses flat filters so it's easy to pop the coffee puck into the compost bin, blast remnants on the (paper or metal) filter in its basket with the vegetable sprayer into that day's graywater bowl, then toss the whole thing on the drying rack. Paper filters for it are almost cheaper than dirt but when they reuse so easily with water I'd already need for plants, why not? Same process if I'm using a metal filter that day - I vary depending on the flavor I want - so I don't have to think before caffeine, the whole routine's on autopilot & takes two aromatic minutes from the point I finish grinding beans. Efficiency is the highest form of beauty, right?