How to put a 1ft piece of a 4x4'' post onto existing one without the connection being visible?
Unless you stain or paint the fence, you will see the joint no matter what since the grain will not line up.

For this type of project I would recommend using a scarf joint. 



This provides a much stronger connection than a simple butt joint. You get an increased glue area and the fastener geometry combined with the scarf will prevent the two pieces from bending apart. (You can use dowels or carriage bolts or screws. Dowels would be the least visible.)

A square scarf joint (as pictured) is pretty easy to make and very strong, but if you really want to hide the glue line as much as possible, you can make a diagonal joint. 



That's a bit more work, though, especially if you have to do a lot of posts. The square scarf will tolerate a little less cutting precision as well.