Feb. 8, 2021
Wearing my parent hat as well as my admissions advisor visor (sorry ... couldn't resist ;-) ), I urge you to take a deep breath, step back, and let your daughter take her junior PSAT in the fall without any further mention of score improvement. If you are already in high gear over PSAT results in grade 10, you risk putting your daughter under significant stress, if she's not there already.
So my advice would be to use the PSAT as it was originally intended ... as a warm-up for the SAT and, especially, as a trial run to see where the weaknesses lie. Then, if you wish, provide some tutoring to prepare for the SAT in the following spring. Since your daughter has already taken the PSAT (twice, no less), you probably already have some sense of where those weaknesses are. But the PSAT is really a test designed for high school juniors, so perhaps her 11th grade results may make up for some of the past deficiencies without additional preparation.
Alternatively, consider the ACT. I think that it can be an easier test when it comes to making improvements. The Science component alone offers many answers to questions right in the questions themselves. The hard part of the ACT Science section isn't the actual material but the time limit. Students typically have to rush to finish. But, with practice at home, most can get the hang of this section and boost scores significantly, once they know how to hone in on correct answers and move through the section efficiently.
As far as scores required for scholarships, it depends on the college involved. Some schools (e.g., the Ivy League and many of the most “elite" liberal arts colleges) don't give any merit scholarships at all, so their grants are entirely tied to financial need and not to test results. However, at these hyper-competitive places, test scores need to be at the top of (or above) the median ranges in order for students without a “hook" (e.g., athletic talent, minority and/or disadvantaged background, any atypical background, VIP or legacy connection) to get in at all. Unfortunately, to find median score ranges these days, you must use a site like the College Board's (see https://www.collegeboard.org/) AND also a “Concordance Chart" to compare results on the new SAT (given March 2016 and beyond) with the old medians. (See https://collegereadiness.collegeboard.org/sat/scores/understanding-scores/sat-score-converter ). All of this conversion craziness is another good reason to defect to the ACT!
In order to wangle some merit aid at a college that offers it, a student's SAT or ACT results usually must be at the top of the median range or, ideally, higher. Applicants with an ACT score in the low 20s or a Math+Critical Reading SAT score of about 1100 (on the old scale) will often get some merit aid at less selective universities ... maybe a couple thousand bucks per year. At the other end of the spectrum, students with 33+ ACT's and 1450+ SAT's may be in the running for full tuition. The score ranges for merit money are as diverse as the colleges and universities themselves. And colleges typically combine test scores, grades, and other achievements in order to determine which students will receive merit money ... and how much. So the cut-off test scores will vary dramatically, depending on the college in question, and scores may play a major role in merit-aid determinations but are rarely the only factor.
When it comes to National Merit Scholarships, these typically go to students whose PSAT's are in the top 1% in the country, give or take. So although your daughter's sophomore scores are nothing to sneeze at, you'd probably have to chain her to a desk until October in order to achieve gains that might put her on National Merit turf. Note also that National Merit is often much ado about nothing. Many colleges don't recognize the program, and some that do award only minimal scholarships for it. A relatively small number of colleges do offer a free ride to National Merit winners, but the best merit scholarships are typically those not affiliated with National Merit. So please don't get out those chains!
Bottom line: "The Dean" recommends no tutoring for the upcoming PSAT. Then you can work on the shaky spots thereafter and/or consider the ACT. From your daughter's scores so far, it sounds like she is a good student, so she ought to have plenty of enviable college options and probably merit-aid options as well. Do help her navigate the maze ahead but also try to ease up on the testing and let her enjoy the high school years that remain and not view them as merely a training ground for college beyond.