So sorry you lost your mature Tulip. The growth from your stump is most likely not growing on their own root, but rather off the cambium ring on the trunk or roots. If you carefully dig down into the soil where they are coming from, you will know if they are growing on their own root. I would expect to see them coming from the top of an existing root near the base of the stump. If they are on their own roots, wait until early next spring to dig and transplant.