We construct an explicit hitting set generator (HSG) for ordered read-once branching programs of width 4. For every length $n$ and every $\varepsilon \in (0,1)$, every width-4 length-$n$ program that accepts more than an $\varepsilon$ fraction of its inputs accepts at least one output of the generator, and the seed length is $$O\big((\log n)^{3/2} \cdot \sqrt{\log\log n} \cdot (1 + \log(1/\varepsilon))\big).$$ For constant $\varepsilon$ this is $O(\log^{3/2} n \cdot \sqrt{\log\log n})$. More generally, it is $o(\log^2 n)$ whenever $\varepsilon \ge 2^{-o(\sqrt{\log n/\log\log n})}$. This is the first explicit generator for width 4 that improves on the $O(\log^2 n)$ seed length of Nisan's generator (Combinatorica 1992).

Our construction first reduces the hitting problem for width-4 programs to the hitting problem for programs with three live states and one rejecting state, following Doron and Hoza (RANDOM 2025). It then simplifies these programs by rounds of pseudorandom restrictions, and hits the simplified programs with a small-bias string that is modified in a small number of positions.