There is no t direction to travel. The infallen observer has a proper time τ and 3 space dimensions. He can move freely along the transverse θ and φ directions, while his radial coordinate r may only decrease, as his τ must always increase.
t is the coordinate time of a stationary observer far away from the black hole, in terms of this time the journey ends when the path freezes at the horizon, so in this frame of reference nothing behind the event horizon ever happens because it already takes an infinite amount of t for the horizon to even form.
To say the infallen observer with proper time τ shall move in the t direction makes as little sense as to demand the outside observer with proper time t to move back or forth in the τ direction. That is neither his time- nor one of his space coordinates, it's only the time coordinate of an observer he is no longer causally connected with.
It is a mathematical artefact that the time of an outside observer runs backwards again after it took an eternity for the testparticle to even reach the horizon, see MTW, Fig. 32.1
Update after comments:
The time dilation of the test particle from the perspective of the far away observer is
$$ (1) \ \ \ \ \rm \frac{d t}{d \tau} = \frac{1}{\sqrt{1-v^2} \sqrt{1-2/r}} $$
(which would be negative behind the horizon at $r<2$, where the local velocity relative to the singularity $\rm v>c$, since $1/i/i=-1$), and the time dilation of the far away observer from the perspective of the test particle
$$ (2) \ \ \ \ \rm \frac{d \tau}{d t} = \frac{\sqrt{1-2/r}}{\sqrt{1-v^2}} $$
(which would be positive even behind the horizon, since $i/i=+1$). Nevertheless, equation $(1)$ is only valid down to $\rm r=2$ and up to $\rm t=\infty$, since it is not physically meaningful to travel to the end of time and back again, as we have seen in the MTW reference (that's what motivated Eddington and Finkelstein to construct their advanced time coordinate which remains valid even behind the horizon).