But apparently this contraption was not
fired at warp, because it originated from a starbase that claimed it couldn't send the passenger aboard a ship because none were available... A starbase supposedly can't travel at warp!
Also, there are several examples in TNG of these small probes being fired at crawling speed, and soon providing information from across interstellar distances (most notoriously "Inner Light"). They could well have the ability to accelerate to warp on their own, then.
OTOH, the probes seen achieving such feats are of a special design that does not look like a photon torpedo. This design could also be somewhat larger than a photon torpedo (if we forget about some ambiguous "full scale" prop work in "The Enemy") - and it has two semicircular wings that curiously resemble Vulcan warp engines...
Possibly the Class 8 Probe is a multistage affair: the Vulcan warp engine accelerates the larger whole to high speeds, at which point the instrumented, photorp-sized payload separates and continues at warp. Or then this particular Class 8 Probe was launched by a booster, available at the starbase, while those used by starships are not boosted that way; only the larger probes with the Vulcan-style drive (usually called Class 1) are capable of high warp when ship-launched.
All this musing becomes irrelevant, though, if Starfleet indeed has miniature warp engines. And we must remember that photon torpedoes often travel at warp, despite leaving the launcher of the starship at very low speed (unless the launch is seen in slow motion, perhaps?).