One of the most popular email services, Gmail keeps track of all your emails with threaded conversations, tags, and Google-powered search to find any message you need.
Email approved hire packets and record a receipt HR can check.
Your new hire packets go out after HR approval, with a receipt showing what you sent and which forms still need signatures.
Send final employment agreements for signature.
Your final agreements reach the signers you list, and you get a reply with the invitation details in the original email thread.