funding - Are there grants and fellowships for PhD students past the first year, but before candidacy?

Effectively as per the questions title, are there grants and fellowships for PhD students past the first year, but before candidacy? From what I have seen, most are intended for first year students (i.e. those with no more than twelve months of study), after candidacy has been achieved, or are finishing fellowships for the last semester of work. For example, the Hertz Foundation says the following:

We generally do not award fellowships to students who are already beyond their first year of graduate study except in cases of "exceptional leverage". Such awards are very rare—only three have been made in the past 10 years. (Leverage here means what difference the award of the Hertz Fellowship is likely to make in the kind, quality, and/or personal creativity of the student's graduate research.)

Which seems to be a fairly common theme. As such, I'm looking for answers in two parts:

1. Are there open grants and fellowships for PhD students past the first year, but before candidacy?


2. Are there field specific ones (computational social science or computational science & engineering) that may apply?