You can be available to the newcomer now. It could be simple as walking up to them afterwards, shaking their hand with a warm smile, pat them on the shoulder, and tell them to keep coming back.
After all, they are walking into a strange environment probably a basket full of emotions. Sometimes simple gestures go along and make them feel at home as much as humanly possible.
If you want to do more, then do more. Give them your phone number. Ask them if they want to grab some coffee sometime. Whatever feels comfortable. You seem like you are in a decent shape for this and would have something to offer if they took you up on it. Run it past your sponsor on what sort of boundaries you should draw regarding newcomers.
If you are looking to sponsor someone, then I would at least wait at least another year, if not two. Again, talk to your sponsor about this.