The term Progressive had been in America for decades at this point. Historians even refer to the period 1896-1916 as the "Progressive Era", when both parties had serious progressive wings. This was the great era of reform.

However, over this period Progressive has been used in a number of different ways. It was the mantel under which three of the most significant third party candidates of the early 20th century ran: Theodore Roosevelt in 1912 (27.4% of the national vote), Robert La Follette in 1924 (16.6%), and Henry A. Wallace in 1948 (2.4%, but many people had expected him to get higher).

In 1955, likely a small town mayor would be thinking something along the intersection of La Follette's vision and Wallace's vision. Indeed, many of the state parties from La Follette's time that had limped along for years were reinvigorated or revived around Wallace's run.

Interestingly, in mid-century American politics, politics were not very partisan as we understand them. There were liberal Republicans (slightly later, these would be called Rockefeller Republicans) especially in the Northeast and there were conservative Democrats especially in the South. La Follette was originally Republican but drew a lot of support from left wing Democrats and Wallace was originally a Democrat (he had been FDR's VP before Truman and was Commerce Secretary under both FDR and Truman) but perhaps his most prominent supporter was a former Republican Congressman from New York, Vito Marcantonio. (Fun side note: Though originally Republican and indeed a protege of Fiorello LaGuardia, Marcantonio hilariously used to run in, and win, the Republican, Democratic, and American Labor Party primaries in his district until they made doing so illegal specifically to target him.)

These two parties were, essentially, the left-wing of American politics and came out, in both cases, of not only the Progressive Era progressive tradition aimed at reform (especially urban reform) but also a specific "Farmer-Laborer" tradition, which sought to have rural and urban workers make common cause to solve their economic grievances. (The two traditions are also not at odds with one-another—the precursor to the organized Farmer-Labor Party in the US was called "the Conference for Progressive Political Action".) This tradition is mostly forgotten today, except that Minnesota's state Democratic Party is officially the Minnesota Democratic–Farmer–Labor Party as in 1944 the state Democratic Party fused with state Farmer-Labor Party.

So what would progressive mean? Both of these two progressive parties were on the left-most edge of the mainstream American politics. Many of the issues we associate with the left today: not just economic issues like support for the New Deal, unions, higher wages for all workers, a national healthcare system, expanded welfare, and public or cooperative ownership of certain industries (Wallace emphasized energy while La Follette had a more sweeping vision), but also perhaps social issues like desegregation and women's rights and being generally anti-war and pro-civil liberties. How that would be translated to the local level might vary—some might emphasize the pro-worker measures, some might emphasize the minority and women's rights aspects, and some might just emphasize the "good government" and government services aspects (one part of the American socialist movement embraced what they called "sewer socialism", emphasizing services offered rather than socialist theory). There was always some hope of creating an American Labor Party on the lines of the British Labour Party (which displaced the Liberal Party as the left-wing rival to the Conservative Party).

In 1955, there might be one element that also comes into play here: socialism. The progressive were generally "pinks", not "reds". That is, while they didn't embrace explicit Marxist-Socialist theory, they also didn't reject Socialist or Communist support. For instance, Eugene Debs ran for President on the Socialist ticket in 1904, 1908, 1912, and 1920. However, in 1924, the Socialist Convention chose to endorse La Follette's Progressive Party rather endorse a candidate of their own (in some states, La Follette ran under the Socialist Party line on the ballot). Likewise, the Wallace campaign decided not to reject Communist support (they decided it would hypocritical considering they were fighting the emerging Second Red Scare) and indeed American Communists were enthusiastic about Wallace's campaign. Wallace was explicitly opposed to the emerging Cold War divisions. So, could being a "progressive" have been a way to support a "socialism that dares not speak its name" in 1955? I don't know. I haven't seen that movie since I was a child so I can't speak to the exact context of this.

So, in short, progressive wasn't necessarily a partisan indicator meaning "Republican" or "Democrat" as it would become in the 1980's when American liberals shied away from the word "liberals". It was still something that existed in both parties, even though the progressive Republicans would be snuffed out by the Conservative Revolution kickstarted by Barry Goldwater and whose culmination was Ronald Reagan. It would, though, generally be associated with all the "left wing" positions of day, generally being pro-worker, anti-big business, pro-civil liberties, pro-public services (from healthcare and welfare to sewer systems and swimming pools) and potentially pro-minority (including women's) rights.