First, the speed of light, as measured by a local observer, is always the same, that is $c$.
To correctly pose your problem, you have to use a light-cone version of Kruskal Szekeres coordinates
The metric is given by :
$ds^2 = F(r) dU dV + r^2 d\Omega^2$, where F(r) is some function of r
The black hole interior is given by $U > 0$ and $V > 0$
The outgoing null geodesics are given by $U = Cte$ (with $V$ increasing).
The ingoing null geodesics are given by $V= Cte$. (with $U$ increasing).
The future horizon is given by $U = 0$ and $V > 0$.
The future singularity is in $UV = 1$ with $U > 0$ and $V > 0$.
So, now, imagine, you are in the interior of the black hole, that is $U > 0$ and $V > 0$, you are sending a outgoing radial light signal, but this signal is at $U= Cte$, so the variable $U$ stays $>0$. But the future horizon is $U = 0$ and $V >0$. So your outgoing signal never reachs the (future) horizon, because the value $U=0$ is never reached.
It is better to draw a litte diagram with the coordinates U and V orthogonal, With upwardly directed axes, and with U and V making an angle or 45 degrees from the vertical.
To complete the schema, you have also :
The past singularity is in $UV = 1$ with $U < 0$ and $V < 0$.
The past horizon is given by $V = 0$ and $U < 0$.