choice of transfer deeds

My parents own a small undeveloped lot in California. My father recently passed away. My mother would like to gift the lot to me, her daughter. The county recorder has instructed us to first do an affidavit of death to remove my fathers name from the deed, together with a preliminary change of ownership form to put her name on the deed. Once this is done, we are told to file for another premilinary change of ownership to add my name, the dauther to the deed along with my mother. My mother would like to 'gift' the lot to me which has been gifted down through prior generations. Our question is, would it be smarter to do a quitclaim deed to ad me as the daughter, or follow the advice of the county recorder to file for a preminiary change of ownership. Whatever we do, we do not want the lot to be 'reassessed' which would cause an annual property tax increase. Because the lot was first gifted from my grandparents to my parents, there has never been an annual property tax increase and we'd like to keep it that way. Thank you for your advice.