Is it permissible to perform supererogatory prayers between the first call to prayer (which is after zawal) and the second call to prayer on Friday, knowing that the Sunnah reported was before zawal during the time of the Prophet, peace and blessings be upon him?

Voluntary Prayers for One Awaiting Friday Prayer:

There is no prohibition against praying at midday on Friday for one awaiting Friday prayer. It is encouraged to perform voluntary prayers while awaiting Friday prayer, before the Imam ascends the pulpit. The established view is that the Hadith of Salman al-Farisi (may Allah be pleased with him) particularizes the general Hadith prohibiting prayer at midday, thus permitting absolute voluntary prayer until the Imam ascends the pulpit. This absolute voluntary prayer is not limited to the time before the sun declines (zawal). So, if the Imam delays ascending the pulpit until after zawal, there is no prohibition against performing voluntary prayers at that time. There is no established regular Sunnah prayer before the Friday prayer. Rather, it is recommended to perform absolute voluntary prayers without specifying a number, and without limiting it to the time of zawal, and this continues until the Imam ascends the pulpit.