The Villain gets +1 Combat skill during the single Fight Round
Actually, the rules do mention Evil Elders outside the Showdown section. Specifically, on page 15.
Evil Elders - When a Town Elder is shown to be evil, they cease to be a Town Elder and instead become an Evil Elder. The card is flipped over, any Secrets they had are discarded, and they are repositioned next to the Villain’s Record Sheet. Evil Elders give the Villain +1 Combat and have one Wound.
This bonus is applied to the Villain's Combat, regardless of whether this is a Showdown fight or not.
Fighting the Villain - Occasionally a hero will have to engage in a single Fight Round with the Villain due to a Mystery Card or Minion Chart result. These Fights are NOT considered a Showdown and consequently, the Villain cannot actually be hurt. In these cases, a single Fight Round should be played out between the Hero and the Villain as described above (the Villain rolls Fight Dice equal to its Combat).