Plot hole: For a convicted murderer who violated her parole and assaulted her parole officer while escaping custody, Ashley Judd moves around the country and even boards airplanes with little to no problems.
Suggested correction: She was simply careful. There's constant manhunts for much more serious felons and parolees on the lam who seem capable of moving around without getting caught.
How did she keep the gun if she flew across the country?
She could have checked the gun in her luggage. Most countries and airlines allow that.