In my opinion, fit has the largest impact on how much scholarships you will be offered. Private schools seem to have more latitude in offering scholarships, although I'm sure there are some exceptions.
I narrowed down my options for both undergrad and graduate school to a small list. I recommend 1-2 high-end "reach" targets (e.g. Ivies), 2-3 schools you have a strong inclination to go to based on their culture, program offered, and recent focus and initiatives the institution most cares about (e.g. best fit schools you'd really enjoy). Have 2-3 "safeties" to apply to where you know you'd get in cheaply without much effort required as back-ups.
I'd then focus on having top-notch, quality applications to each of the schools you'd most like to attend. After thoroughly researching the culture of the schools and found ones with programs which fit your interests and place graduates into jobs you'd be interested in after graduation... be obsessive about using essays to explain 1) your background, 2) why this background makes the school a great fit for you, and 3) what you intend to use your education for post-graduation.
This shows maturity and seems to be an indication of what admissions boards want to see - someone who enjoys their experience at the school and go on to become successful leaders in a given field. Since you have the grades and scores, focus on fit and making a great impression.
This is an approach that worked for me personally. However, it's now been six years since I graduated... get views from some other folks as well to sense-check my recommendation.
Taylor recommends the following next steps:
- Narrow and bucket your list of schools to apply to
- Knock essays and interviews out of the park
- Wait and watch the scholarship offers pour in