Views contextual filter - Contains search with taxonomy field
As per my understanding exposed filter works for this requirement. So maybe you can try combining contextual filter with exposed filter.

You can use the Views Filter Harmonizer module for that. Here is a quote about it (from its project page):


  Views Filter Harmonizer solves an operational foible with the Views
  module regarding filtering.
  
  Normally, when a Views field has been assigned both an exposed filter
  and a contextual filter, then both filters get applied to the View's
  results set -- always. This is not configurable behaviour. It's
  hard-coded.
  
  As a consequence, contradictory search clauses easily occur, causing a
  counter-intuitive user experience with some White Screens of
  Disillusionment. Since the contextual filter sets the "bounds" for the
  page, any filtering by the visitor through the exposed filter on that
  page can only narrow down the results set, not broaden it. In other
  words the exposed filter appears not to be working properly.
  
  This small module allows you to have a contextual filter argument
  apply only when there is no corresponding regular (exposed) filter
  value present.
  
  This is most common on initial page load. After that the contextual
  filters will silently retreat and let the exposed filter values as
  selected by the visitor do their thing without interfering.
  
  This means that you can take advantage of friendly URLs using
  contextual arguments instead of ugly and verbose query parameters. Put
  the URLs anywhere on your site, in emails and in newsletters to take
  visitors to tailored Views pages. Having arrived there, visitors may
  then further filter these pages to their heart's content using the
  View's regular filters you exposed. And they won't experience any odd
  behaviour.