Just going on the information in your question it is difficult to know about your circumstances. Under the Federal Copyright Act, the creator of the work owns it unless it is a work made for hire. But, there must be a written agreement for them to own the work. The other exception which does not apply in your case is if an employee does the design for an employer the employer owns it.
Carefully read all documents forming the agreement between yourself and the graphic design company. They may have retained all rights, but, it’s possible to get what’s called an assignment so you own the logo and have authority to sell it to your new company without running afoul of the law.
Do not do this yourself. Retain an intellectual property attorney to save yourself from potential liability and to save money in the long run.
If you check the U.S. Copyright Act at 17 U.S.C 101 you’ll see why a professional should do this.
Andrew Myers of Derry has law offices in Derry and North Andover. He is a member of the American Association for Justice and the New Hampshire Trial Lawyers Association. Send questions to email@example.com.