Answer: Yes, some annual flowers do better in cool or warm seasons, and you may even see them labeled as "fall-winter" or "early-flowering" annuals.
Some of my favorite cool-season annuals include calendula, bachelor's button, pansies and violas, nasturiums, sweet peas, petunias, larkspur, geranium, scarlet flax, Bells of Ireland, stock, snapdragons, alyssum, dusty miller, poppies and dianthus. If you plant them early enough, or buy bedding plants from a greenhouse, you'll see blossoms in spring, and they'll have another flush of bloom as the weather cools in fall.
Warm-season annuals include sunflowers, tithonia (Mexican sunflower), zinnia, salvia, verbena, hollyhocks, butterfly weed, coreposis, cosmos, gaillardia, black-eyed Susan, coneflower, lisianthus, and vinca. I hope this info helps!
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