Seems you have a case of locker room filter. What we do after a hike, and also in town on longer trips every week, is dissasemble the whole thing and stick it all, hoses and everything into a bucket(or sink) with a gallon of water and a strong dose of bleach (a tablespoon or two) and let it sit for a while. If the cartridge wasn’t too dirty and it still passed water put it back together and then pump the bleach solution through it for a few quarts worth. Now switch to clean water and pump a gallon through it to wash out the bleach. Rinse everything well. Take it all apart again and air dry it. Leave the filter out until its dry. Don’t forget to lube the o-rings on the pump plunger and the filter with silicone paste. You should have gotten a small 1/4 oz tube with the new filter. Save a dab for later. This maintenance should stop the cooties from growing. Some of the funk is from your hands if the outside of the pump smells like arm pit.:x
Bushwhack