Expected overshoot sum when uniforms first pass one

Let $X_1,X_2,\dots$ be independent $\text{Unif}(0,1)$ variables, set $S_n=X_1+\cdots+X_n$, and let $N_1=\min\{n: S_n>1\}$ be the first index at which the running total exceeds $1$. The value $\mathbb{E}[S_{N_1}]$ has the form $ce$ with $c$ rational. Determine $c$.

$c=\tfrac12$. A classic result is that on average $e$ uniforms are needed to pass $1$, i.e. $\mathbb{E}[N_1]=e$. Because $N_1$ is a stopping time, Wald's identity says the expected stopped sum equals that mean count times the mean of one uniform, $e\cdot\tfrac12$.