Suppose $X$ is a separable infinite dimensional Banach space, $E$ a finite dimensional subspace which is $c$-complemented in $X$. Is the following true?

For any $\varepsilon>0$, one can find $x\notin E$ such that $\mathrm{span}\{E,x\}$ is $(c+\varepsilon)$-complemented in $X$.

This is obviously true in Hilbert spaces, but not sure in general Banach spaces. Would reflexivity make a difference?