A highly developed sense of harmonic relation - it helps if you can instantly transpose a short phrase to any given key.The ability to play multiple accompanimental figurations, ranging from oom-pah-pah waltz type accompaniment to scales or arpeggios against any given phrase.The ability to create joining passages (typically a brief cadenza or recitative passage will work well here).
The ability to take any phrase and return to it at will at a later point, to be able to create variations on it, to manipulate the rhythm of it, to maintain the rhythm but change the notes (ie use it as a motivic cell), and the ability to expand it at will, or to go sideways from it (eg the movitic cell example above). Also the ability to create answering phrases of the cuff. All of these will facilitate creating structure, if what you want is an improvisation which has points of commonality within it.