Generalization of homotopy lifting property To paraphrase the normal homotopy lifting property, it states that given a covering map $\pi:E\to X$, an interval $I=[0,1],$ a homotopy $f:Y\times I\to X,$ and a lift $\widetilde{f}_0:Y\to E$ of $f_0:Y\to X$ (where $f_0(y)=f(y,0)$) so that $\pi\circ \widetilde{f}_0=f_0,$ there exists a homotopy $\widetilde{f}:Y\times I \to E$ lifting $f$ so that $\pi\circ \widetilde{f}=f$.
I am wondering if there is a generalization of this property where the interval is replaced by a path connected space $T$ with a basepoint $t_0,$ so that given a "homotopy" $f:Y\times T\to X$ and a lift $\widetilde{f}_{t_0}:Y\to E$ of $f_{t_0}:Y\to X$, there exists a "homotopy" $\widetilde{f}:Y\times T \to E$ that lifts $f:Y\times T\to X$.

Path-connected won't work : otherwise take $Y=*$, then you would be claiming that any continuous function $T\to X$ lifts to $E$, which is known to be wrong.
However, if you require $T$ to be simply-connected and locally path-connected, then the answer will be yes, at least under the usual hypotheses of covering space theory.
Namely, here's a nice generalization of what you're looking for :

Assume $X$ is a nice space. Let $f: Y\to X$ be a continuous map with $Y$ path-connected and locally path-connected, $f(y)=x$ and $x_0\in E$ a lift of $x$. Then there is a lift to $E$ sending $y$ to $x_0$ if and only if the image $f_*\pi_1(Y,y)$ is contained in the image $\pi_*\pi_1(E,x_0)$; and if it is, the lift is unique.

If $T$ is simply-connected, then the image of $\pi_1(Y\times T)$ will be the same as that of $Y$ and so your assumption implies that there is a unique lift on $Y\times T$ having the desired value on $(y,t_0)$. Note that composing with the inclusion at $t_0$, $Y\to Y\times T$ provides a lift of $f_{t_0}$, so it must be $\tilde f_{t_0}$ by uniqueness.
