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This is not yet an answer, but a first step to avoid the rather unwieldy Irwin-Hall distribution. From the characteristic function, I find that $$P_c\equiv\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]=\frac{1}{2}-\frac{1}{2\pi}\int_{-\infty}^\infty dt\,\frac{1}{(it)^{n+1}}\left(e^{it(1-c)}-e^{-ict}\right)^n$$ or, if we write $c=1/2+\delta$, $$P_{1/2+\delta}=\frac{1}{2}+\frac{i}{2\pi}\int_{-\infty}^\infty dt\,e^{-2int\delta}\,\frac{\sin^n t}{t^{n+1}}$$