This is a topic where the schools of jurisprudence have different positions. The Shafi'i and Hanbali schools permit performing multiple Umrahs in one trip, citing the hadith where Nabi (shallallahu alaihi wa sallam) instructed Aisha's brother Abdur-Rahman to take her to Taneem to enter Ihram for a second Umrah during the Farewell Haji. The Hanafi school generally permits it as well.
The Maliki school and scholars like Ibn Taymiyyah consider performing multiple Umrahs in one trip to be disliked (makruh). Their reasoning is that the companions of Nabi, despite living in or frequently visiting Mekkah, did not repeatedly perform Umrah. They advise that voluntary Tawaf around the Kabah is more meritorious for someone already in Mekkah than going out to Taneem to re-enter for another Umrah.
If you choose to perform a second Umrah, the process is: exit the Haram boundaries to the nearest Miqat point for Mekkah residents, which is Masjid Aisha (also called Masjid Taneem), approximately 7 km from the Haram. Enter Ihram there with the intention for Umrah, then return to the Haram and perform the complete Umrah rites (Tawaf, sa'i, and hair cutting/shaving). There is no fixed waiting period between Umrahs. Many jamaah haji perform a second Umrah on behalf of a deceased relative or someone unable to perform it themselves.