The ones that fall off the plants immediately walk their way back up. Try spraying the soil and trays with the soap solution, then going around squishing with soapy hands and/or knocking the aphids off with a soft bristle paint brush into the soapy bubbles, then finish with spraying with water to rinse off the leaves. This routine could take the place of watering.
Are they all in trays? I hold individual containers sideways and brush the aphids off into soapy water. Soft watercolor brush really helps to get into tender new leaves.
If they are solidly in trays and won't come out, I've seen videos of commercial growers holding the trays upside down and DIPPING the seedlings.
Take steps to keep ants away from the seedlings.
Oh, one more thing -- find some spiders and put them among the seedlings. Walking/hunting spiders are more effective than ones that sit in a web.
ETA -- I had another idea. You must be setting them out to harden off by now, right? When you do, surround them with flowers that attract beneficial insects like Sweet Alyssum (I think they would be selling them in full bloom by now in 6-packs, etc.).
Today, I was going to check one of my tomatoes already planted in the ground which had been stressed by the frosty nights spent under the tub and was attacked by aphids. Last time I looked, it had lost much of its dark green color and was infested by dark green aphids that had sucked the green out of it. But I'd noticed it was greening back up nicely.
When I peered in close, I realized there were hoverflies walking around on the leaves. I pulled back my hands that I had ready to turn over leaves and squish and left the Garden Patrol to their task.
