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