How can I sign on with the incoming unit to stay for another rotation in Kosovo?

First you need to talk to your Commander/Command and will need to get a release from them. Then you need to contact the National Guard Rep who is at Bondsteel, you can check with the Mayor's cell there on Bondsteel to get the number /e-mail. This is going to be the best way to go as he/she will have a good handle on the rotation and can contact in incoming unit and know the needs. Most units are filled before hand.

Why do you want to stay for another rotation and how long is the current rotation.

First you need to talk to the incoming unit, and make sure they want you, and have a slot for you in your MOS. Once you find out that, it's just a matter of talking to your unit and the incoming unit to do the paperwork to amend your orders, transfer you over, and such.