To manage your lawn effectively before overseeding, it's recommended to let your grass grow until it reaches a height of about 4-6 inches before cutting. Cut only one-third (1/3) of the height of the grass blades when you do mow. Since your lawn is currently at 3 inches, you should allow it to grow to around 4-5 inches before considering your next cut.

A rule of thumb is to wait until your grass reaches this height, which typically takes about 1-2 weeks under ideal conditions, depending on factors like weather, sunlight, and moisture. However, if your lawn is healthy and growing well, you might be able to mow sooner if you follow the one-third rule and maintain overall grass health during overseeding.

Please make sure to prepare your lawn correctly before overseeding as well, ensuring good soil contact and ideal conditions for seed germination.
https://yardandgarden.extension.iastate.edu/how-to/overseeding-lawn
https://extension.umn.edu/lawn-care/renovating-lawn-quality-and-sustainability See lawn reunovation section.