Well, let's throw another bit into the game. From "Metamorphosis" we also have the entity traveling at warp speed "staying right" with the shuttle.
SPOCK: Heading directly toward us at warp speed.
KIRK: Staying right with us. Sensor readings, Mister Spock.
And later on, they're looking for antimatter trail from the shuttle:
UHURA: Mister Scott. Computer central reports that we're coming up on the co-ordinates of the last established position of the shuttlecraft.
SCOTT: Thank you, Lieutenant.
SULU: Steady. No, Mister Scott, bearing three ten mark thirty five just cleared. No antimatter residue.
SCOTT: All scanners, spherical sweep. Range, maximum. They'll have to pick it up.
UHURA: If the shuttlecraft powered away, Mister Scott, but if it were just towed?
SCOTT: There'd still be traces of residual matter floating around, Lieutenant.
SULU: Bearing two ten mark forty. Strong particle concentration. We're on it, Mister Scott.
I'd like to think that the shuttlecraft represents earlier versions of TOS FTL systems before the introduction of Lithium/Dilithium that gave them "regenerative power". But also that in TOS, a variety of engines were FTL capable ranging from the warp engine, hyperdrive and ion engine. It doesn't match with TNG and later productions, but I don't put the two continuities (universes) as one so their technology trees could be fairly different, IMHO.
Edit: @Timo - the catch with "residual" fuel being used at the end is that Scotty was pretty absolute in saying "We have no fuel". It looked like to me that during that adaptation process, new fuel was generated, IMO.