Yes, travel to Masjid Aisha (Taneem) to enter ihram for Umre after completing all Hac obligations — this has a basis in the Sunnah from the hadith of Aisha.
Yes, performing an additional Umre after completing Hac is permissible and has a basis in the Sunnah. Hz. Peygamber (sallallahu aleyhi ve sellem) instructed Aisha (may Allah be pleased with her) to go to Taneem with her brother Abdur-Rahman to enter ihram for Umre after she completed her Hac, as narrated in Sahih Bukhari and Muslim.
To perform this Umre, you must first complete all your Hac obligations, including Tavaf al-Ifadah, Say, and the stoning of the Cemerat. 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 Umre, then return to Mescid-i Haram for tavaf and sa'i. Some scholars consider this extra Umre merely permissible while others view it as sunnet. It is best done before Tavaf al-Wida (farewell tavaf) if you plan to perform one.
Source: Sahih al-Bukhari, Kitab al-Umre; Sahih Muslim, Kitab al-Hac