Ingmar Guandique was indicted this month on counts including obstructing justice after authorities said he conveyed the threats from jail.
Guandique's lawyer entered a not-guilty plea for him to the new charges during an arraignment Tuesday in D.C. Superior Court.
Guandique has already pleaded not guilty to first-degree murder and other counts filed in April over Levy's death.
Levy had just completed an internship with the U.S. Bureau of Prisons when the 24-year-old disappeared after leaving her apartment in jogging clothes in May 2001. Her body was found a year later in Rock Creek Park.
Guandique has been serving a 10-year sentence for separate assaults in Rock Creek Park.