Was Star Trek’s “Time Warp” ever developed past episode one of the TOS?

In "The Cage" and "Menagerie" Pike specifically tells the crew "Time Warp Factor" instead of "warp factor". Possibly it was shortened to plain warp factor by the series.

In "The Cage" and "Menagerie" Lt. Tyler also says that the "time barrier" has been broken and new ships are faster. Thus it would seem that the "space warp" discovered by Cochrane ("Metamorphosis") was used in the original warp drive but newer warp drive also has a "time Warp".

Thus in "Where NO Man Has Gone Before" the "space warp" engines of the Enterprise were disabled and the ship was traveling much slower, but it apparently was still traveling faster than light since it traveled several light days in less than one day of ship time, apparently still being able to use the "time warp".

I repeat that in "Where NO Man Has Gone Before" the "space Warp" was disabled but it seems certain that the Enterprise was still able to use the "time warp" generators to either ONE) travel several light days at faster than light speeds or to TWO) travel several light days at slower than light speeds but slow down the passage of time aboard so that less than a single day passed aboard.

This "time warp" ability explains how doing something new with the engines sent the Enterprise back in time in "The Naked Time".