Here are some quality schools with 20-30% admit rates (or higher) that give strong financial aid to international students. For most of these schools, your chances may be less than 20-30%, but anyway. The schools I list below give average aid of at least (about…) 50% of the total cost to attend:
Vassar
Oberlin
Reed
Bates
U of Richmond
Kenyon
Macalester
Lafayette
U of Miami
Trinity College
Connecticut College
Franklin & Marshall
Union
St. Olaf
Denison
Dickinson
Skidmore
Bard
Lawrence
Beloit
St. Lawrence
The article is three years old, but I imagine most (if not all) of these schools still give strong aid to internationals:
http://thecollegematchmaker.com/65-colleges-give-generous-aid-international-students/