OP,
If you have a 4.0 and a 1560 SAT, there are indeed LACs that will offer half tuition. Most LACs top-out at half-tuition (that’s their highest merit award), but a few offer full tuition scholarships (Agnes Scott (if female), Knox College, Wooster (only one or two, I believe), Hiram College, and a few others).
If you are aiming for full tuition scholarships, a Google search will produce a pretty comprehensive single list. I believe that list is here on CC as well (maybe even pinned?). A good rule of thumb is that you will get your highest merit awards from the schools where are are a significant cut above the average admitted student.
But, yeah, there will be half-tuition merit for a student with your stats, at many schools probably.