Introducing a new team member is a big change for everyone in the team, so be conscious of keeping them front of mind throughout the process. If you have the need for a new person because someone has resigned, make sure the reasons for this are clearly communicated. If they have left the company, avoid rumours and discontent starting by addressing front on why they have left.
If there are issues, this is a good opportunity to resolve them with the team. Conversely, if they have taken a promotion either within the company or elsewhere, what a great opportunity to profile to the team the career opportunities available to them.
Remember your team is the best source of recruitment referrals. They should be strong ambassadors for your company and team and should jump at the chance to promote an opportunity to someone in their network.