Can you settle for two outlets?
There's no such thing as a mid-cable "tap".
Any splices must be done inside a junction box. The box must remain accessible forever. You need at least 6" of wire inside the junction box - 9" is better so you have some margin for screw-ups. Therefore aim for at least 18" of slack.
The only workable way is to have TWO boxes set at least 18" apart along the cable run. (or somewhere else if the length allows, e.g. two boxes 12' apart and down 5 feet). Then, you add additional cable to connect the two boxes to restore the original circuit (e.g. a 24' run that goes up 5 feet, over the 12 feet, and down the other 5 feet).
Again, the boxes must be accessible forever. They cannot be covered up with any building material (a cabinet with hinges and a latched door is fine. Easily lifted drop ceilings are also fine. A lift-out false back to a bookcase is debatable since the electrician will never find the darn thing when trying to chase a wire problem.)
It's perfectly acceptable to bring the splice box somewhere visible and just go ahead and make it a receptacle box. You can have receptacles in weird places. (however if they're GFCI receps that need to be TESTed and RESET sometimes, they must be sanely located).