Let's look at the validity of tachyons first. Travelling faster than light, as said by userLTK, would create time travel, which would create the famous paradox, the tachyonic antitelephone. The tachyonic antitelephone is a violation of causality, and thus most likely, tachyons would not exist.
Now, take a look at another assumption. Wikipedia is right almost all times, but sadly, it has some wrong information. Such is with tachyons. It gives us some contradictory statements of tachyons too. It is wrong to assume that tachyons always move faster than light. A tachyon, is a particle with an imaginary number as its rest mass. This, however, does not mean it can travel faster than the speed of light, nor that there's any conflict between their existence and the special theory of relativity.
The main idea here is that the typical intuition we have about particles -- them being billiard ball-like objects -- just fails in the quantum world. It turns out that the correct classical limit for quantum fields in many situations is classical fields rather than point particles, and so you must solve the field equations for a field with imaginary mass and see what happens rather than just naively assume the velocity will turn out to be faster than light.
Leandro M. wrote on physics stack exchange, that
a good way to think about tachyons is to imagine hanging several
pendulums on a clothesline, one after the other. If you disturb one of
them, some amount of force will be transmitted from one pendulum to
the next and you'll see a traveling disturbance on the clothesline.
You'll be able to identify a "speed of light" for this system (which
will really be the speed of sound in the string). Now you can make a
"tachyon" in this system by flipping all the pendulums upside down:
they'll be in a very unstable position, but that's precisely what a
tachyon represents. Nevertheless, there's absolutely no way that you
could send a signal down the clothesline faster than the "speed of
light" in the system, even with this instability.
This means that tachyons don't actually move faster than the speed of light, and your assumption is wrong.
Furthermore, StephenG wrote in a comment that tachyons essentially can't exist in a black hole.
user106422 wrote in the linked question on physics stack exchange that
in a nutshell, creating a tachyon itself would lead to some form of
a spontaneous phase transition in which the tachyonic mode no longer
exists. So from a rigourous QFT + GR setting, I don't think tachyons
exist the way it is portrayed in simpler QM models. So no, tachyons
will not escape the horizon simply because the theory will not allow
for them to exist.
Thus, no, even if they exist, they don't really travel faster than the speed of light, and even if they do, that can't really "travel" through a black hole. So, it is not possible to use tachyons to explore a black hole.