Look at this from another angle. It's always easier to find a job when you already have one. You are more attractive as a candidate when you are already employed. Never jump from the boat you are in until you are close enough to the dock. If you jump too early, you will get wet!
Try to look at your current job as an open interview for your next position. Your next employer may walk in the door at any moment. If you are going above and beyond in your service, they are going to notice. At the very least, your managers and owners of your company will see you adding so much value, they can no longer ignore you. They will have to promote you or risk losing you to a competitor. Now you are in a win-win situation!