Eldritch Horror skill test for multiple combat encounters on same turn
From the base rule book, page 9:


  Combat Encounters
  
  If an investigator is on a space containing one or more Monsters, he
  must resolve a single Combat Encounter against each Monster on that
  space, one at a time, in the order of his choice. Combat Encounters
  are described in detail on page 14.
  
  After an investigator resolves all Combat Encounters, if there are  no
  Monsters in his space, he may resolve another encounter.
  
  The card effect only lasts as long as the combat encounter, and the
  combat encounter lasts only for fighting a singular monster.


The Wither Card states:


  When resolving a Combat Encounter, you may test Lore. If you pass, gain +3 Strength during that encounter. 


Hence, the combat encounter is the process in fighting a singular monster and you can have multiple combat encounters in a singular encounter phase. Wither only lasts for a singular combat encounter (single monster) and gives +3 strength for it. 

You would need to cast the spell again in the next combat encounter if you wanted it's effects again. 

Make sure you flip the card each time you cast the spell to resolve the effect on the back.