Yes, travel to Masjid Aisha (Taneem) to enter ihram for Umrah after completing all Hajj obligations — this has a basis in the Sunnah from the hadith of Aisha.
Yes, performing an additional Umrah after completing Hajj is permissible and has a basis in the Sunnah. The Prophet (peace be upon him) 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 Hajj, as narrated in Sahih Bukhari and Muslim.
To perform this Umrah, you must first complete all your Hajj 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 Masjid al-Haram for tawaf and sa'i. Some scholars consider this extra Umrah merely permissible while others view it as recommended. 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-Hajj