Currently, there are $350$ students in the seventh grade at Winchester Heights Middle School. For the upcoming school year, they are planning to increase the number of seventh graders by $10\%$ .

To find how many students there will be in the upcoming school year, we need to calculate $10\%$ of the current number of seventh graders and add that to the current total.

$10\%$ of $350$ students is $0.10 \times 350 = 35$ students.

Therefore, Winchester Heights Middle School plans to have $350 + 35 = \boxed{385}$ seventh-grade students in the upcoming school year.