The bulbs will 'store' better and easier in the ground than anywhere else - I'd leave them alone. However you should also be aware that most tulips do not return in following years with the same sort of vigor they had with their initial planting. You may get more foliage than flowers. Some people treat them as annuals for that reason, replanting with fresh bulbs each fall.

Thank you gardengal for your insight! I had no idea that they might not come back with the same vigor in following years. A good reason to try out more bulbs each fall perhaps:)?!

For tall returning tulips try the Darwins. 'Daydream' is a great multiplier/naturalizer according to one of the ladies here. Another mentioned her returning Darwins have alternate years of good bloom (the splits have to grow large enough to bloom).

I keep learning more and more:)! Love it. Thank you for your suggestion about the Darwin Daydream Hybrid. Apparently it is also fragrant! I did not know they existed. I will read up on them in preparation for fall.