Yes, travel to Masjid Aisha (Taneem) to enter ihram for Umrah after completing all Haji obligations — this has a basis in the Sunnah from the hadith of Aisha.
Yes, performing an additional Umrah after completing Haji is boleh and has a basis in the Sunnah. Nabi (shallallahu alaihi wa sallam) instructed Aisha (may Allah be pleased with her) to go to Taneem with her brother Abdur-Rahman to enter ihram for Umrah after she completed her Haji, as narrated in Sahih Bukhari and Muslim.
To perform this Umrah, you must first complete all your Haji obligations, including Tawaf al-Ifadah, sa'i, and the stoning of the Jamarat. Then travel to the nearest miqat point outside the Haram boundary — Masjid Aisha at Taneem is the closest (approximately 7 km, easily reached by taxi for about 20-30 SAR). Enter ihram there with the intention for Umrah, then return to Masjidil Haram for tawaf and sa'i. Some scholars consider this extra Umrah merely boleh while others view it as sunnah. It is best done before Tawaf al-Wida (farewell tawaf) if you plan to perform one.
Source: Sahih al-Bukhari, Kitab al-Umrah; Sahih Muslim, Kitab al-Haji