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 the Prophet (peace be upon him) instructed Aisha's brother Abdur-Rahman to take her to Taneem to enter Ihram for a second Umrah during the Farewell Hajj. 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 the Prophet, despite living in or frequently visiting Makkah, did not repeatedly perform Umrah. They advise that voluntary Tawaf around the Ka'bah is more meritorious for someone already in Makkah 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 Makkah 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 pilgrims perform a second Umrah on behalf of a deceased relative or someone unable to perform it themselves.