There are a couple of posts on here for people looking for a current player league that is just beginning. Unfortunately, it would appear that no one is starting any new leagues in that format. All of the new leagues being formed are fictional or historical.
I am in a couple of leagues, and I have yet to get a chance to join an actual current player league in it's infancy. One thing that I have seen however, is that the maturity level in current player leagues seems to be a bit on the...low side. Also I have seen a couple that have finished the 2004 season, and then, a bunch of people quit and then the league is in bad shape.
This is just my
but you may want to consider looking for either historical or fictional. Unless the above is your thing.
One of my favorite leagues has Boston and Seattle open, we are in the 1998 season, and soon it wil be a current player league
http://nahhbl.ootp-online.com/
Our commisioner bigphesta is one of the best and the sims times are pretty reliable.