From an earlier answer of mine

Except for a few isolated cases very early in the movement, the NSDAP never used the word "Nazi" in its official lexicon and its use was actively discouraged by the state. Goebbels did write a pamphlet Der Nazi-Sozi - Fragen und Antworten fuer den Nationalsozialisten in 1932. But his use of the word "Nazi" is limited to the title and Goebbels tends to use NSDAP/NS/National Socialistische conventions inside the text. The term itself was a derogatory term with its origins prior to Hitler's seizure of power. Many NSDAP members often referred to themselves as NaSos (short for National Socialists), and their opponents instead called them Nazis. The word Nazi had a negative connotation in southern Germany because it resembled the Ignaz, which the rough translation is "ignorant" or "blockhead." The word Nazi became quite commonplace among the many political enemies of the NSDAP, more than a few of which ended up in exile after Hitler's rise to power. These emigres would help popularize the word Nazi as they often wrote disparagingly about the direction Germany had taken since 1933.

German written discourse in the Third Reich had a number of means to refer to both the National Socialist political movement and its ideology. Most commonly, writers tended to use "National Socialisten" or "National Socialistische" as either a noun or adjective. Writers would sometimes also use the abbreviation N.S. for space saving purposes (this is quite common when referring to state organizations such as the NS-Frauenschaft (National Socialist Women's League). For the Party itself, the acronym NSDAP was pretty universally used.

Nazi was sometimes used within Germany while Hitler was in power, but seldom openly. Most of the vernacular slang against the regime usually used some variation on Braun (Brown), the official color of the NSDAP uniform (aka Brown Shirts). One grim joke during the dictatorship was that Germany has been renamed Braunschweig, because one half is "Braun" and the other is schweig (silent). Other jokes drew on the color's association with dirt and excrement (eg Why are we now tapeworms? Because we have to live inside the brown and hope to be excreted.). One term for various NSDAP officials was "golden pheasants" both for their bedazzled uniforms and ostentatious living standards during the war.

Such jokes show the somewhat anomalous position the Party had in German's daily lives. While many Germans joined the NSDAP and participated in its institutions, there was an undercurrent of resentment towards NSDAP corruption and incompetence. Moreover, a number of these jokes and aspersions also speak to a public fears that the NSDAP was a transient phenomenon, which became especially salient as the war turned against Germany. Yet many Germans, even the grumblers, were complicit in Nazi crimes and actions. One grim joke common after Stalingrad "enjoy the war, the peace will be terrible," was evidence of the knowledge of war crimes and possible retribution.

To add a little more to the importance of names, during wartime the Soviets frequently referred to the Germans as fascists. This was in part due to the Marxist-Leninist idea that Mussolini and Hitler were part of the same counter-attack by monopoly capital on the rise of the USSR, but also a means to not call attention to the Socialist component of the NSDAP's name. The label "fascist" struck a number of German observers as somewhat odd because the Third Reich made it rather clear that the NSDAP was a distinct political movement quite separate from Mussolini's one. While Germany's state media stressed that Germany and Italy shared common interests, they were not part of a single political ideology and German media use of the word "fascist" was mostly to refer to affairs in Italy.

The answer to your question is a bit more complex than you might initially think.

Nazi was used before the Nationalsozialistische Deutsche Arbeiterpartei (NSDAP) was even formed (renamed from DAP in 1920, which in turn was formed in 1919).

Nazi in its earlier use was a short form derived from the name Ignatz. In southern Germany it was sometimes used as a derogatory phrase for austrians and/or idiots. This dual meaning fit the stereotyping. One dictionary (http://www.etymonline.com/index.php?term=Nazi) indicates it might have been used for austro-hungarian soldiers in general, but I've never come across that myself.

During the 1920s the term shifted towards the usage we are more familliar with now and had negative connotations from the start. Usually members of the NSDAP used the full version Nationalsozialist or the shortened form NaSo to describe themselves. But Nazi is analogous to the term Sozi (short for Sozialist) which was used for members and proponents of the social-democratic SPD. This might have contributed to its increasing usage for members of the NSDAP.

We can observe this shifting meaning by examining the prolific journalist Kurt Tucholsky. In 1922 he (under the pseudonym Peter Panter) published an article titled 'Die Nazis' (http://www.sudelblog.de/?p=376) using the word as a derogatory term for austrian cultural elitists coming to Berlin. He even quips about 'Knüppel-Kunze' (referring to Richard Kunze. Kunze was busy with his own right-wing political activities at that time, but his views were clearly leaning their way (and later on he became a member of parliament for the NSDAP). But the NSDAP was small fry in 1923 so Tucholsky used the word as a slur for stupid austrians, despite the fact that this right-wing politician would without a doubt be dubbed a 'Nazi' today.

By 1930 the NSDAP had risen to be a major political party. Tucholsky (under the pseudonym Theobald Tiger) published the poem Deutschland erwache! (Wake up Germany! http://www.zeno.org/Literatur/M/Tucholsky,+Kurt/Werke/1930/Deutschland+erwache!). In it we can see that Nazi has fully turned into a shorthand for the NSDAP and its proponents. E.g. daß der Nazi dein Todesurteil spricht / Deutschland, fühlst du das nicht? (that the Nazi is issueing your death sentence / Germany do you not feel that?)

So in a sense 'Nazi' was associated with something bad even before the NSDAP became a major political party. In any case, it certainly was used as a derogatory term for Hitler's party before Hitler got appointed as chancellor.