I have several that qualify: Strawberry Candy (I'm sure everyone has this one:), Jedi Tequila Sunrise, Modern Design, Head Turner, Cherub's Kiss (started blooming May 23rd and is still blooming!), Evening Enchantment, South Seas, Midnight Raider, All Fired Up, and a few NoIDs.
I live in SC, so I think our growing conditions are optimum for daylilies to have ample time to have instant rebloom and what I refer to as a regular rebloom, which is a few weeks after the first scapes have faded away. Some will keep producing scapes all summer.
Here are four of my NOIDs that send up instant scapes:
Pictures 1, 2, & 3 are the same plant. Then 4 and 6 are a morning/afternoon of the same plant. Pictures 5 and 7 are obviously different plants.
I also had a couple of plants to have multiple scapes from the same fan this year, something I had not ever noticed on any daylily before--Jolyene Nichole & Leebea Orange Crush were the two I remember off the top of my head. LOC is in two beds, and both clumps had this going on.