If you require $C = P^1$ then it's probably not possible except for very small values of $r$. If you don't care about $C$, then here is something that might work.
Suppose $E$ is given by $y^2=x^3+ax+b$ and $P_i=(x_i,y_i),i=1,ldots,r$ is
a basis for the Mordell-Weil group. Let $C$ be the curve given by the system
of equations $u_i^2 = (t^i+x_i)^3 + a(t^i+x_i) + b + t, i=1,ldots,r$ in
$t,u_1,ldots,u_r$ and $S$ be the family $y^2 = x^3 + ax + b+t$ pulled back to
$C$. So above $t=0$, $C$ has a point with $u_i=y_i$ and and the fiber of $S$ above this point is $E$. Also $C$ is defined so that there are sections of $S$ with $x$-coordinate $x=t^i+x_i$ and I bet they are independent. Finally the family is non isotrivial if $a ne 0$. If $a=0$ adjust the construction is an obvious way.
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