The simplest answer is yes. Yes, you can ask that your employee sign a noncompete agreement at any point prior to or during their employment. However, in order for your noncompete agreement to hold up in the court system you should pay particular attention to the details while creating it.
A noncompete agreement is essentially a promise made by your employee, that upon the dissolution of your employer-employee relationship, he or she will not disclose previous employer information to a future, competing employer. Contrary to previous belief, a noncompete agreement cannot be used to limit an individual's ability to earn a living. They are used as a way to protect a business's goodwill and trade secrets.
If and when a noncompete is challenged in court there are several aspects of the agreement that are looked over when testing its validity. These things, such as the reason for its use, and scope of time it is in place, must be considered reasonable and key to the protection of the business. In reference to when an employer can request a noncompete, the idea of consideration must be present at the time it was signed. In this sense consideration is something given to the employee in return for the promise made in the agreement. If a noncompete was requested prior to the individual's employment, the court will assume the individual agreed to the noncompete in exchange for the job. If a noncompete is requested after employment there must be something else offered to the employee in exchange for the agreement, whether a promotion or raise, a noncompete must always be supported by this consideration.
A noncompete agreement can be a valuable resource to employers. However, if not created properly its validity may come into question. An employment law attorney may be able to help create a bulletproof noncompete that will keep your business's best interests safe.