Turning one lonely observation into the sample mean

You have i.i.d. readings $X_1, \dots, X_n$ from a Normal distribution with unknown mean $\mu$ and known variance $\sigma^2$. A lazy analyst estimates $\mu$ using only the first reading, $T = X_1$. It is unbiased but wastes almost all the data.

Use the Rao-Blackwell construction to improve $T$ by conditioning on the sufficient statistic $S = \sum_i X_i$, and show exactly how much the variance drops.