Yes, travel to Masjid Aisha (Taneem) to enter احرام for عمرہ after completing all حج obligations — this has a basis in the Sunnah from the hadith of Aisha.
Yes, performing an additional عمرہ after completing حج is permissible and has a basis in the Sunnah. نبی کریم (صلی اللہ علیہ وسلم) instructed Aisha (may Allah be pleased with her) to go to Taneem with her brother Abdur-Rahman to enter احرام for عمرہ after she completed her حج, as narrated in Sahih Bukhari and Muslim.
To perform this عمرہ, you must first complete all your حج obligations, including طواف al-Ifadah, سعی, and the stoning of the جمرات. 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 احرام there with the intention for عمرہ, then return to مسجد الحرام for طواف and sa'i. Some scholars consider this extra عمرہ merely permissible while others view it as مستحب. It is best done before طواف al-Wida (farewell طواف) if you plan to perform one.
Source: Sahih al-Bukhari, Kitab al-عمرہ; Sahih Muslim, Kitab al-حج