It’s been awhile since I’ve used their strips, but as I remember their strips have a male (pins) end and a female (socket) end. The receiver/controller has a short pigtail with a female end. You’ve plugged a strip in using the male end of the strip and cut that strip in half leaving you with a strip with a cut end and a female end.
To make that cut piece useful, you may use one of two methods: 1) unplug your strip from the receiver/controller and install a “T” or “Y” shaped adapter (the adapter has one male and two female connections) and then you replug in your first piece into one leg of the “T”. Now to be able to use your second cut off strip, you need a male to male adapter (a small piece with pins out both sides). Plug the male to male adapter into the “T”. Now plug the scrap strip’s female onto the male to male adapter. 2) Buy a second receiver/controller, get/buy a male to male adapter, plug it into receiver/controller and plug scrap strip (female end) into male to male adapter.
So as explained, if I remember correctly, you need to buy some stuff to make your scrap piece useful. It is unlikely you will find what you need wherever you bought the strip (Lowe’s?), but there is an OSRAM Lightify adapter kit. It includes splitters (“T” or “Y”) and extension cables and male to male pin adapters. Nearly everything you need to make your scrap piece functional and or make uncut pieces into two or three strips.
If you add a second receiver/controller, you will still need a male to male adapter.
Things to remember, number of pins. OSRAM strip come in two flavors, four pins or five pins. Make sure you buy everything with the right number of pins.
Remember the strips and all connections are polarities sensitive. Most OSRAM stuff has a little tiny arrow on the strips and "T"s and extension cables. The arrows must match (point at each other) or nothing works.
Amazon sells everything you may need
SYLVANIA Smart+ Flexible Strip Connectors for Smart Home (14 pieces) - Connected LED Light Strips, 73826 (Formerly LIGHTIFY), Works with Amazon Alexa
Sylvania Smart Home