This is extremely difficult to answer because it reaches into counterfactual territory and because the term Holocaust is so deeply tied to gas chambers and industrialized genocide.

The leaders of the Third Reich had no problem with openly declaring war against the Jews as it fitted into the narrative of "race darwinism". Actions against Jews like the boycott or the November Pogroms of 1938 were visible to everyone (and only few non-jewish Germans openly opposed them) and were portrayed as "Volkswillen" (the people's wish) in propaganda and media.

Then again, SS members participating in the mass killings in death camps were prohibited from talking about their actions by death penalty. But this, as far as the sources tell, only applied to the actual death camps and not the mass shootings in Eastern Europe. We have lots of sources of soldiers telling their families and friends about those elements of the Holocaust when they went home for vacation.

So, my educated guess is that they wouldn't have publicly announced the Holocaust as in telling everyone about Auschwitz, Zyklon B and crematories, but they would've tried to paint the death of millions of Jews as the result of an ongoing war the Jews had declared against Germany - which would be easier to convince everyone of when victims are shot and not round up, gassed and burned in industrial proportions.