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

Question

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?

Answer 1

Yes. Several of the fellowship and scholarship mentioned in a previous question (Options other than the NSF for PhD level grants?) are open to non-first year graduate students.

For example, The EPA STAR Graduate Fellowship, The Ford Foundation through the National Academies, NSA Fellowships, Google Research, and The Data Incubator all allow people to apply past the first year.

The NSA, Google Research, and Data Incubators Fellowships all target computer science. "The Ford Foundation seeks to increase the diversity of the nation’s college and university faculties by increasing their ethnic and racial diversity, maximize the educational benefits of diversity, and increase the number of professors who can and will use diversity as a resource for enriching the education of all students." The EPA Fellowships target environmental science, which could include computational science applied to studying the environment.

Edit note November 2019: The EPA no longer offers their fellowship program.

Answer 2

One example is the National Physical Science Consortium, emphasis mine:

Eligibility:

NPSC welcomes applications from any qualified U.S. citizen who has the ability to pursue graduate work at an NPSC member institution. Applicants at any stage of their graduate program may apply, as long as they will be available to accept two summers of paid internship. Those who already possess a doctoral degree are ineligible.