Hi there, I am also in the process of getting my new DNI after renewing my temporary work visa for a second time. Unfortunately, my DNI and wallet were stolen during the renewal process, so I have to get a new one with the Registro Nacional de las Personas. You can check with the Registro, and they have information available here and you can get a turno online: http://www.nuevodni.gov.ar/
Otherwise, I am pretty sure you can continue to use the old DNI, as they usually just place a stamp in your passport for your official visa renewal. I just used my previous DNI after the first renewal, and when I traveled, I just presented the DNI along with the visa renewal page in my passport. Hope this helps!