Yes, travel to Masjid Aisha (Taneem) to enter ইহরাম for উমরাহ after completing all হজ্জ obligations — this has a basis in the Sunnah from the hadith of Aisha.
Yes, performing an additional উমরাহ after completing হজ্জ is permissible and has a basis in the Sunnah. নবী করীম (সাল্লাল্লাহু আলাইহি ওয়া সাল্লাম) instructed Aisha (may Allah be pleased with her) to go to Taneem with her brother Abdur-Rahman to enter ইহরাম for উমরাহ after she completed her হজ্জ, as narrated in Sahih Bukhari and Muslim.
To perform this উমরাহ, you must first complete all your হজ্জ obligations, including তাওয়াফ al-Ifadah, সাঈ, and the stoning of the জামারাত. 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 ইহরাম there with the intention for উমরাহ, then return to মসজিদুল হারাম for তাওয়াফ and sa'i. Some scholars consider this extra উমরাহ merely permissible while others view it as মুস্তাহাব. It is best done before তাওয়াফ al-Wida (farewell তাওয়াফ) if you plan to perform one.
Source: Sahih al-Bukhari, Kitab al-উমরাহ; Sahih Muslim, Kitab al-হজ্জ