Why didn’t the Weimar Republic use higher denominations during hyperinflation?

They did. The government issued notes in ever higher values, such as this note for 20 billion Marks.

Issuing higher numbered denominations was not a solution to the currency crisis, but rather a symptom of the deep problem of German post-WWI money policies. The hyperinflation of 1923 involved a currency called the Papiermark. This was a form of currency that had no real concrete backing such as gold to prop up its value. The complex fiscal situation of the Weimar Republic in which both the legacy wartime fiscal decisions as well as Allied demands from the Versailles Treaty and the German government's reaction to them led to Papiermark losing its actual value. This led to the period of hyperinflation in which the government printed out Papiermarks of insane values, such as the 5 billion Mark or the 20 billion Mark linked above. This created a period of acute political crisis as savings were wiped out and radical outliers to the postwar political scene gained in prominence. Among the latter groups was a hitherto marginal Bavarian far right group, the NSDAP which launched its failed takeover of Bavaria, the Beer Hall Putsch, in this period.

The way out of this "witch's sabbath," as one German economic historian put it, was through a new currency to be put into circulation. The Papiermark was replaced by the Rentenmark and the older currency could be exchanged at a rate of 1 Rentenmark for 1 Trillion (!) Papiermark. The Rentenmark was backed by a hypothetical mortgage on German agricultural and industrial property and limited to 3.2 billion Rentenmarks. The Rentenmark was only an interim currency; it could not be circulated out of Germany and its main purpose was to drive out the Papiermark and restore fiscal sanity. This took a little bit of time, but by 1924, the Reichsmark (RM) replaced the Rentenmark as the national currency of Germany.

The RM remained the main currency of Germany through the Republic and Third Reich. The Nazis' economic policies which prioritized rearmament at insane levels for a peacetime economy potentially could have resurrected the specter of hyperinflation, but Hitler was very keen on placating public expectations for a sound currency. Thus the dictatorship adapted various schemes such as MEFO bills (a shell game with the Reichsbank to pay for rearmament), wage freezes, and other schemes that gave the appearance of stability. The reality was the RM had become hostage to German militarism and expansionism once the balloon went up in 1939. It held value only if Germany was winning the war. The Allied occupation continued to print RM in their occupation zones after 1945, but some (although not all) of the same factors of the early 1920s were at play in the post-1945 period and the solution to the danger of a valueless currency was replacement by a new one. Cooperation between the Anglo-American zones and the emerging West German government led to the creation of the Deutsche Mark in 1948 while the Soviet controlled Eastern zone followed suit with what became popularly known as the Ost-Mark.