Is there a way to create a Membership Status for memberships that have recently been renewed?
Well, it's not really possible to have a status for the Renew state because the Start event used for status rules is based on one of the following date :


start date : this date is updated only when there is a gap between the last end date and the new renewal date... if there is no gap, which is the most frequent case, there will be no update
end date : updated any time the membership is successfully renewed / mainly used for "grace" / "expired" status
member since : this date is not updated by the system at all so it is used for new membership


So, without development, the only way i can think of having this kind of status is to manually update the start date every time you renew (which probably defeat the purpose).

Then, you can define :


Start Event to start date
Start Event Adjustment to be 0 (start at the renewal date)
End Event to be start date again
End Event Adjustment to be 1 month or the period you'd like it to lasts
Current Membership should be checked because obviously, membership with those status are valid


Please put the new Renew rule above the Current status by changing the Order because CiviCRM test the rules one by one and stop at the first matching item.