I suspect the OP is using knotless, tapered leaders. If that's the case, then out of the package the leader is essentially one piece, just tapered down to whatever size tippet the package is marked as. So to a newbie it basically seems like you're cutting your leader as you remove the flies...you are in a way because it's all one continuous piece.
If that's the case, as the others have mentioned, once you've snipped off a foot or two of the fine "tippet" end, then just tie a new piece of tippet of the same diameter on, and tie your flies on to that...repeat until you've worked up into the thicker taper of the leader.
One other suggestion...you could go to a furled/braided leader with a loop style connection on the tippet end. You just tie new tippet onto the loop as you use it up. They're more expensive ($10 a piece or so), but I can often get an entire season out of one leader that way. You're only cutting/removing/replacing tippet using this system, the leader stays in tact the whole time.
Bottom line though...you gotta snip the flies off and take the curled/knotted piece of tippet with it. You need new, straight tippet to get a good new knot, and the tippet will be weaker than its advertised breaking strength if you retire with the curled/knotted section from the previous knot.