I ran a regression $Y \sim X_1 + X_2 + .. X_n$. I find out what one regressor , $X_1$'s performance depends on another variable $t$ (not in the regression). So basically if I bucket by $t$, within each bucket of $t$, I plot the correlation of $X_1$ and $Y$. I can see that when $t$ is higher, the correlation between $X_1$ and $Y$ is very obviously higher.
One example is $t$ is time of the day. $X_1$ has higher correlation with $Y$ as time goes by during the day, for may reason. But $t$ itself doesn't predict $Y$ at all. My $t$ and $X_1$ can be positive or negative.
So I want to take advantage of this phenomenon. I scale $X_1' = X_1 * (1+t)$. Basically when $t>0$, I increase the magnitude of $X_1$, and vice versa. I rescale $t$ to be within $(0,1)$ or even smaller intervals. Then I run regression $Y \sim X_1' + X_2 + .. X_n$
Much to my surprise, this adjustment doesn't improve my regression at all, based on all the regression stats.
How do I improve my regression based on this observation?