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