Easy one 1st-replace the printer with one that prints photos "edge-to-edge". All of the big names make them and they are competitive in price.
The only other option, that I can think of, is to crop the image greater in size than you want and then cut the borders off. For example: use 5X7. The problem with this method is that it is hard to keep the edges clean: even with a paper cutter. You will get a full 4X6 image but lots of busy work. Based on your photo software and the printer, you might have to play around with the sizing.
For example: I have some software that allows "custom" size cropping as well as the fixed sizes. You might have this as well. Might be frustrating.
So my printer prints 4 x 6 photos. I've been using a cannon sd550. The problem is that I like to frame my photos to fit the camera's screen and when I go to print I get big ugly white bordsers on my 4x6 prints because the ratio on the camera and the printer do not match.