From my use of center pull skeins this should not be happening. Some of the low-end pull skeins (notably Red Heart) tell you to take the outside tail out at one end and then to look for the pull end from the other end of the skein. Lots of other brands don't have this information, but I think there is something to it.
Some skeins have both ends tucked in a little in the ends somewhere. Some have the outside end tucked in/or loose, and the pull end you have to "dive" for. I think you should always locate the outside thread, and make sure it is not tucked in anywhere, before you go looking for the pull part. If you find that outside end tucked in anywhere at one end, choose the other end to go "diving" for the pull end.
If you can't associate the outside tail with either end you just have to guess which end to look in for the center pull. I try one end and if I can't find anything fairly easily I try the other end. Center pulls are not always deep inside (if not, they are usually right at the end, even sticking out a bit) but if they are, I try to go to the very center of things and pull something there. I do get a bit of excess sometimes with that first pull, but I have never had recurring issues all through the skein (if this is an unintentional lie, it is at least not so common, because I can't remember it).
If you get that outside end somehow caught in the skein you can have troubles further along though, I'm not talking about if that happens.