it depends on what you are trying to accomplish and a bit on the size of the company.
In general, when speaking of a small company (i.e. 50 staff members) I would not rebuild the infrastructure, because the effort is relatively high.
However, Microsoft recommends using a name which is never used publicly with a routable suffix (i.e. namespace.com).
For Active Directory, I know the recommendation to use a sub domain of this domain (i.e. ads.namespace.com) but I did not find the related Microsoft article.
So in your case using a sub domain like "ad." would be fine.