Seasons like Spring and Fall, with moderate temps and more moisture, are ideal times to plant grass seed (and just about anything else).
Spring is the second-best option for cool-season grasses, especially if you missed the fall window. However, be mindful of summer heat approaching, which can stress newly planted grass. You’ll need to water frequently to help the seedlings establish.
For warm-season grasses (like Bermuda, Zoysia, or Bahia), late spring to early summer is ideal. These grasses thrive in warmer temperatures and need the heat to germinate properly. Planting in late spring allows the seeds to take advantage of the warm weather, giving them the best chance to grow and develop during the hot summer months.
In all cases, planting when the soil temperature is around 55°F-65°F for cool-season grasses or 65°F-70°F for warm-season grasses ensures the best germination and growth.
This article has some great tips on best practices for planting grass seed.