The politician confirmed on Monday morning that the package contained a 9mm bullet and a sympathy card with his name and picture.
It was discovered at the Royal Mail sorting office at Mallusk, Co Antrim.
Mr McGlone said those behind the sinister threat were "faceless fascists" who would not deter him from continuing to represent his constituents.
"This is obviously not something that is a pleasant experience, for me or my famil."
He also paid tribute to staff at the sorting office and the PSNI for their "professionalism and vigilance."
Mr McGlone said he was "acutely aware "of other elected representatives -from the SDLP and other parties- experiencing similar threats and attacks on property.
He continued, saying that they "have come through this and I will come through this stronger. Democracy will come through this stronger."
Those who sent it will not put me off doing my job - that is representing the people of Mid Ulster
Alliance Party Chief Whip Stewart Dickson condemned those responsible for sending the bullet to Mr McGlone and offered support on behalf of his party.
"Any threat or attack against an elected representative is an attack on democracy. Every politician should be able to work on behalf of their constituents without the fear of violence or intimidation," the east Antrim MLA said.
"We have seen a number of threats against politicians from several political parties in recent weeks as well as attacks against their homes and offices, including my own. This violence must stop now.
"This threat is unacceptable and must be unequivocally condemned by all."
On Sunday, it emerged that the Belfast home of SDLP councillor Claire Hanna had been attacked.
Shots were fired at the front windows and door from what is thought to have been a high powered ball bearing gun.
It is believed the incident happened sometime between Thursday night and Friday morning, however, the councillor, her husband and baby daughter were not at home at the time.
Ms Hanna, the Chair of the East Belfast Policing and Community Safety Partnership, told UTV that the attack disgusted her and showed utter contempt for her young family.
"It's a violation of our security.
"My husband and I have a young daughter and obviously, her safety and security is our principle concern in life and this incident has very much shaken us."