With college acceptance letters out, high school seniors and their families now turn to what has become an annual ritual in the application process: financial aid appeals.

A little strategic haggling before the May 1 student decision deadline can cut $5,000 to $10,000 more a year off the tuition bill, says Joel Peck, a certified public accountant in New York who advises families on college admissions.

Those who are new to the process might not be aware they do not have to pay the base sticker price, which for private nonprofit colleges averaged $33,480 this year, according to the College Board.

To attract top candidates, many colleges are willing to shave money off tuition, even beyond what they have already offered in financial aid, on a case-by-case basis. “They call it merit aid to deflect from the fact that it is discretionary,” Mr. Peck said.

Appeals may surge this year because of recent changes in the federal student aid application that allow families to file their paperwork earlier and base it on tax returns from two years ago. The older information leaves families enough wiggle room to appeal on the grounds that the college didn’t base its financial aid award on the most up-to-date financial circumstances.

And appeals may gain more urgency in the future if the sharp cuts in student aid programs proposed in President Trump’s budget become a reality.

Private nonprofit colleges are the most likely to bargain with accepted students, especially ones with strong grades and test scores that will help improve the school’s rankings, college advisers said.

Mr. Peck says the next few weeks before the May 1 notification deadline are important because as families select their schools, they will free up scholarship and aid money at the schools they do not choose, which will have to be reallocated to other applicants. That is money that is up for grabs.

For the 2015-16 academic year, about 88 percent of first-time full-time private college freshmen and 77 percent of all private school undergraduates were awarded half the cost of tuition and fees, according to the National Association of College and University Business Officers.

That trend is expected to hold up for 2016-17, according to Ken Redd, the association’s director of research, who is busy completing this year’s survey to be released in May. The practice of offering discounts on tuition — which is practiced by all kinds of private colleges, from nationally recognized schools to regional and smaller institutions — has gone up every year since the financial crisis in 2008, Mr. Redd said.

But the financial disclosures required of families can be complicated, and some schools ask for more personal information beyond the common federal form everyone uses. Some families turn to advisers to guide them through admissions. Mr. Peck, for example, charges up to $1,300 for a package of services that runs from the application through the financial aid appeal. He also recently began selling a $297 video about college financial aid.

The number of full-time independent education advisers has exploded in the last decade, to 8,000 in 2015, from 1,500 in 2005, according to the Independent Educational Consultants Association, and about twice as many people dabble in it as a side job, said Mark Sklarow, the association’s chief executive.

There is no official licensing for these advisers, who charge a median fee for a package of college advising services of $4,600. Many will do financial aid appeals on a per-hour basis, Mr. Sklarow said. The typical client is a child attending a large public high school in an affluent suburb who is planning to attend a private college.

College acceptance letters and financial aid awards land around the same time, which, during the standard admissions timeline, is in the spring. Advisers have several suggestions for families planning to appeal; the first one is, lose the attitude.

“You don’t want to bargain like Cuba Gooding Jr.’s character in ‘Jerry Maguire,’ ” said Kalman Chany, an adviser in New York who charges $1,950 for his complete package of college services, and advises on financial aid appeals for $850 to $1,250. “Don’t go in and say, ‘Show me the money!’ ”

Students who are entering their freshman year and have multiple admission offers tend to have the most leverage at the bargaining table. That is because most successful appeals are argued on the basis that the school did not have the best information about the applicant’s true financial circumstances, or that a rival school has made a better offer.

A student could be eligible for a tuition discount because he or she falls into a specific demographic group, like being the child of a graduate, having a certain religious affiliation or being eligible through work. A college with declining enrollment might also be more willing to shave down the tuition bill to fill spots.

But a college might also consider a specific skill or talent as worthy of a tuition discount to coax the student to enroll. Say a student plays tuba in the marching band, and a college has eight spots for tuba players in its band, with three about to graduate. Being qualified to take one of those spots gives the applicant a selling point.

“If the admissions people know they want a particular kid, there will be a note in the file and the financial aid office is going to work with that family,” said Ronald Ramsdell, the founder of the Minneapolis-based College Aid Counseling Services, who charges $1,150.

Families that can document an unusual circumstance that is not reflected on the Free Application for Federal Student Aid, or Fafsa, they filed at the start of the application process have a strong argument to appeal for more financial aid. Some reasons could include a bonus that was only paid once but listed on the old tax form, or excessive medical bills, a dislocation because of a natural disaster or a recent job loss.

Being able to provide solid proof is an important part of a successful appeal, either in the form of those medical bills or the competing offer from another school.

Financial aid awards based on a family’s need can come in the form of grants that don’t have to be repaid, plus loans or work-study programs. Merit aid, on the other hand, is often used to entice certain students. “The better students get the bigger aid packages,” Mr. Chany said. “Ask yourself, how do you stack up?”

Timing is also important, counselors say, to avoid looking desperate and to maximize leverage. Families should wait a few days after receiving the letters offering of admission and financial aid to start their appeal, and should wait to make their final decision as close to the May 1 decision date as possible.

Correction: April 14, 2017

An earlier version of this article misstated what Joel Peck, a certified public accountant in New York who advises families on college admissions, charges for services from the application through the financial aid appeal. It is up to $1,300, not up to $1,500. The article also misstated the price that Kalman Chany, an adviser in New York, charges for his complete package of college services. It is $1,950, not $1,900. The article also misspelled, on second reference, the surname of the chief executive of the Independent Educational Consultants Association. He is Mark Sklarow, not Skarlow.