SAT measure beginning with 8th grade: https://collegereadiness.collegeboard.org/psat-8-9

And, of course, the existing PSAT/NMSQT, and the PSAT/10:
https://collegereadiness.collegeboard.org/psat-nmsqt-psat-10

ACT measure for 3rd to 10th grade: http://www.discoveractaspire.org/assessments/periodic/

The ACT measure includes a version that is more similar to MAP in its use for districts and schools, in terms of repeated measures during the school year. This would, of course, have measures of math and critical reading.

Scholastic also publishes the SMI for math, but I don't have any direct experience with it, nor have I had as much positive feedback on it from other professionals.

...pronounced like the long vowel and first letter of the alphabet...