I doubt the new seal is leaking, as you probably still just have some of the white spouge left in the tranny. In the past, when I have lost the water pump seal, I have used a light solvent to rinse the tranny, and then change the oil a couple of times after brief running. Just put and a quart of light solvent, kick the engine over several times, and then drain it out. Repeat again if the solvent is really ugly. Leave the top and bottom oil plugs out for a day or so before adding oil, and the solvent will flash and evaporate. After a day or so, add and remove oil a couple of times, and you should be set.
If your seal is leaking again, yes, you can change the seal without pulling the cultch cover, but it is tough. You will need a good dental tool with a hook to reach in and grab the seal, and you might even need two dental tools tools to apply evan pressure as you pull the seal out. Once the seal is out just a little, you can grab it with a pair of pliers and finish the removal.
Good luck.
Rick