I know there are avoid notes for the diatonic scale and the melodic minor scale. I am not familiar with exotic scale avoid notes (if they really exist in jazz music), for example, there is the enigmatic scale, hungarian scale and the double harmonic scale. Do exotic scales have avoid notes in jazz music (contemporary or bebop) or are there any artists in jazz music that intend to use the idea of exotic scale avoid notes?
The definition of an avoid note is independent of the type of scale you use. What you need to define an avoid note is a (chord) scale and its related chord. If you define the basic chord of a scale as the seventh chord constructed in thirds starting from the root then an avoid note is
[T]he pitch or pitches of a chord scale which are not used harmonically because they will destabilize the sound of the chord.
(B. Nettles, R. Graf, The Chord Scale Theory & Jazz Harmony).
Usually non-chord tones a half-step above a chord-tone are perceived as destabilizing, so they are the ones considered avoid notes.
E.g., for the enigmatic scale (with root C)
the basic chord would be
C E G# B or Cmaj7(#5). The only avoid note is Db because it is a half-step above the chord tone C.
The double harmonic scale
is full of avoid notes: Db, F, Ab are all avoid notes, if we define Cmaj7 as the basic chord of this scale.
In this way you can derive avoid notes for any scale. However, the question remains what the consequence is of a note being an avoid note. It is avoided harmonically, but many pieces using exotic scales do not use standard four-part chords. Sometimes you just have a drone or a simple chord, where the concept of avoid note loses its importance.
I think it's important to understand the concept of avoid note, but never forget that there is no implied rule as to how to use any note. As usually, let your ears decide.
If you want to read more about avoid notes, check out this article.