If it's seasonal, it most likely is pollen (possibly mold, but more likely pollen in the spring).

Washing your face after a ride will help - as you've been doing. Also wash your eyes, either with an eye wash cup you can get in the drugstore, or just by streaming sterile saline into your eyes.

Pollens will build up on your eyelashes and face even when you're not riding, so it's a good practice to rinse your face first thing in the morning and last thing before you go to sleep - towel off your hair with a damp towel before you go to sleep (or in the morning, if you usually wash your hair at night) - and change your pillowcase often, even daily if it's really bad.

Ask your ophthalmologist if there are drops you can't use. You could also try treating your allergies systemically with OTC antihistamines, but if you're not having other symptoms besides mild conjunctivitis, that's probably overkill.