We give new pseudorandom generators for \emph{regular} read-once branching programs of small width.
A branching program is regular if the in-degree of every vertex in it is (0 or) 2.
For every width d and length n,
our pseudorandom generator uses a seed of length O((\log d + \log\log n + \log(1/\epsilon))\log n)
to produce n bits that cannot be distinguished from
a uniformly random string by any regular width d length n
read-once branching program, except with probability \epsilon.

We also give a result for general read-once branching programs, in the case that there are
no vertices that are reached with small probability. We show that if a (possibly non-regular) branching program of length n and width d has the property that every vertex in the program is traversed with probability at least \gamma on a uniformly random input, then the error of the generator above is at most 2 \epsilon/\gamma^2.