Your name and fame (or infamy) spreads like wildfire through any town you pass through. One random encounter that happened while traveling to the current town (at the DM's discretion) starts a rumor about you that blows out of proportion. You have disadvantage on persuasion and deception checks to dispel the rumors, and advantage on persuasion or deception checks to goad it further.

"Did you hear the rumors? I heard that they killed the monster in one blow!"

"I heard that they didn't even touch the monster and it still died!"