General:
* Ad-lib
* Extemporize
* Wing it
* Make do
* MacGyver
* Scrounge
* Resourceful
* Invent
* Create
* Adapt
More specific:
* Compose (for music)
* Concoct (for a plan or recipe)
* Fabricate (for a story)
* Compose (for a speech)
* Craft (for a solution)
The best synonym for "improvise" will depend on the specific situation and the nuance you want to convey.