Praise be to Allah.
There is no basis for this action in the Sunnah. Although the Sahaabah were keen to do good, they did not do ‘umrah after Hajj, and they were the best of generations. Rather that is mentioned in a specific case with regard to ‘Aa’ishah the Mother of the Believers (may Allah be pleased with her), as she had entered ihram for ‘umrah, then she got her menses before reaching Makkah. So the Prophet (blessings and peace of Allah be upon him) told her to enter ihram for Hajj so that her pilgrimage would be qiraan, and he said to her: “Your tawaaf around the House and your saa’i between al-Safa and al-Marwah will suffice for your Hajj and ‘umrah.” When Hajj was over, she insisted that the Messenger of Allah (blessings and peace of Allah be upon him) let her do ‘umrah to make up for the ‘umrah that she had turned into qiraan. He gave her permission and told her brother ‘Abd al-Rahmaan ibn Abi Bakr to take her out of the sanctuary, so he took her outside the boundary of the sanctuary and she did ‘umrah.
If the same thing happens as happened to ‘Aa’ishah (may Allah be pleased with her) and the woman wants to do ‘umrah, in that case we say: There is nothing wrong with a woman doing ‘umrah as ‘Aa’ishah the Mother of the Believers (may Allah be pleased with her) did when the Prophet (blessings and peace of Allah be upon him) told her to. What indicates that this is not something prescribed is that ‘Abd al-Rahmaan ibn Abi Bakr (may Allah be pleased with him), who was with his sister, did not enter ihram for ‘umrah, either on the basis of his own thinking or the permission of the Messenger (blessings and peace of Allah be upon him). If this were something that is prescribed, he (may Allah be pleased with him) would have done ‘umrah, because that is something that would have been easy for him as he had gone out with his sister. To sum up, what some pilgrims do, as mentioned in the question, is something for which there is no basis in the Sunnah.
Yes, if we assume that it would difficult for some pilgrims to come back to Makkah on another occasion, and they did ifraad, then in this case it is a matter of necessity so after Hajj they my do ‘umrah in order to fulfil the obligation of ‘umrah – because ‘umrah is obligatory according to the most correct scholarly opinion. In that case he may go out to al-Tan’eem or anywhere else that is outside the boundary of the sanctuary, and enter ihram for ‘umrah from there, then do tawaaf and saa’i, and shave his head or cut his hair. End quote.
Majmoo’ Fataawa Ibn ‘Uthaymeen, 22/36