For me, the problem with going the speed of light is that you are then outside of time. Once you were How would you slow back down when you got where you wanted?
So there is the other side if the speed of light, but you can't get there from here, you have to go somewhere else first. And the 'somewhere else' is a place where all material objects have infinite mass.
Theoretically, the only way to get 'superluminal' would be to translate yourself via CPT reversal, or some other translation mechanism that turns "you" into something that is already "tachyonic antimatter". But that is in some sense not travel, just some sort of information re-encoding.
And it may happen spontaneously all the time. (Yes, I am going where I always go with time questions, time either equals or does not equal entropy accumulation.) If entropy does not accumulate in a perfectly linear way throughout the universe, then you occasionally go back in time, but you cannot experience the reversal, because memory is an exothermic process.
So to my mind, the way to get somewhere yesterday is not to pass the speed of light, it is to manipulate entangled states. If time really is subjective, and flows both ways, you should be able to make something in the past that must match a given state in the future or keep some other event un-collapsed forever. Since the odds of the latter are so low, a copy of you would have appeared there, then, to prevent it from needing to happen.
We think we know how to set up quantum states that can only be observed if they meet a given set of conditions. This is the basis of quantum computing: if the bizarre combination of factors that formulate the problem were not met, the computer would have to stay 'uncollapsed' for way too long. So just stating the question forces a solution to happen relatively quickly.
If time really has no implicit direction (or if any kind of paradox prevention is automatic) then creating a causal loop constitutes causation. So stating a problem, and locking it away unobserved means that its outcome is not determined until we observe it. I think we could, if fiendishly clever, force the conditions to be that the solution should match what we are going to put into a given box later.
Then we put you in the box. And you are the unobserved solution, which means you exist shortly after the problem was stated. But now you are observed. The causal loop constitutes a cause of you being instantly duplicated at an earlier time at a given location, which may be arbitrarily far away from where you are now.
Of course, we cannot do that kind of thing now, and you may only ever be able to manage that state at the level of an individual quantum particle. But if you could do it to something with history, it would prove that time's arrow is not a physical reality, but an aspect of perception.