What do you mean by "it sputters"? Does it sound like the engine is struggling to get power once it hits second? If it is, then richen the High-Speed needle (the big brass one) counter-clockwise 1/4 of a turn, or just 90 degrees CCW.

If it was too rich (getting too much fuel) you would probably already know this and it might not even be able to get into second.

If you richen it and it seems to perform a little better but doesn't hit second, maybe this is why it was (if it is the case) struggling for power hitting 2nd. The 2nd gear is set too high and because of this, the engine is trying to rev way too high and the only way it could do it is by running too lean. Sure it might be able to reach a higher RPM, but it would take away some power as well and so it would find it hard to keep running in second.

That's what I think is happening.