  In shift-symmetric Horndeski theories, a static and spherically symmetric
black hole can support linearly time-dependent scalar hair. However, it was
shown that such a solution generically suffers from ghost or gradient
instability in the vicinity of the horizon. In the present paper, we explore
the possibility to avoid the instability, and present a new example of theory
and its black hole solution with a linearly time-dependent scalar
configuration. We also discuss the stability of solutions with static scalar
hair for a special case where nonminimal derivative coupling to the Einstein
tensor appears.
