Generally, the history we are taught is, “the Depression started, then FDR responded with the New Deal.” But Hoover was less than a year into his term when the market crashed.
I can't comment on the role of the government in the crisis but I believe I can speak on the Federal Reserve's response to the crisis, which I've studied through studying monetary policy. I believe the Fed's actions are the most significant when discussing the Great Depression. So much so that Ben Bernanke, who would go on to become head of the Fed, apologised on behalf of the Fed for the crisis in 2002. I am no means an expert on this topic, there's definitely other users here who will be able to add more substantial comments.
At the time, the Fed's central body had less power than it would today. For a national decision to be made, each state's Fed would have to agree, which made responding to the crisis more difficult. An example of this was the issue of the Fed's role as the lender of last resort. When banks began to fail due to depositors withdrawing their cash in huge amounts (a bank run), the Fed's governors disagreed on what was the best approach. At a high level, one side wanted to lend to the banks, the other side didn't want to interfere. The latter side felt that allowing weak institutions to fail would allow a stronger economy to progress. One of such advocates was Herbert Hoover’s secretary of treasury, Andrew Mellon.
One case the Fed initially took the wrong course of action is their interest rates. At the time, the US dollar was still on the gold standard. Policymakers wanted to prevent gold leaving the US to protect the value of the US dollar. To do this, they kept interest rates high and didn’t engage in open market operations (e.g. quantitative easing). Banks had no reserves from deposits due to the run on the banks and borrowing/lending fell. This meant money supply fell considerably, meaning businesses had restricted access to capital which contributed to the fall in aggregate demand, meaning that consumption and spending fell, harming economic growth.
The Fed’s initial response was proven to be incorrect when they began engaging in open market operations to increase money supply, which aided the recovery of the economy from 1933 onward combined with the measures of the new administration.
Sources:
https://www.federalreservehistory.org/essays/great_depression
Mishkin F. (2009), The Economics of Money Banking and Financial Markets, Ninth Edition.
This is the textbook my lecturer used and recommended us to read, there was a ton of other readings which I don't have access to currently.
I can comment a bit on the personality difference between Hoover and FDR and how this affected the way they approached the Great Depression.
The fascinating thing about Hoover is that before the Great Depression he was known for being this kind of embodiment of the American Dream as well as a man who could solve problems. He was a poor orphan who attended Stanford, became a mining engineer, and, through his career in the mining industry, became a millionaire by the age of 40. He went into public service and during World War I managed relief efforts to Belgium (which has streets named after him, and there's also a statue in his home state of Iowa that Belgium donated after the war as thanks) and he headed the United States Food Administration (i.e., convincing people to ration their foodstuffs) with great success.
As Secretary of Commerce under Harding and Coolidge he was known as something of a dynamo as he not only rallied for businesses to cooperate and created standards of goods, quality, prices, etc., but he also called a conference to save the Chesapeake Bay, managed relief efforts for the Great Mississippi Flood of 1927, and regulated radio, aviation, and television. He was the easily the favorite to run for president in 1928. Hoover had all this public acclaim, and the American economy was still prospering—he even stated in his nomination speech that the United States was “nearing the final triumph over poverty.” His image and the prosperous United States were linked.
Then eight months into his presidency the stock market crashes. He doesn’t make no response, but it can be argued that he didn’t do enough, or couldn’t think outside the box, in part because he didn't think the government should have that much power to assist people.
Hoover tried:
Encouraging relief though private charities and channels – these were very quickly overwhelmed.
Meeting with big business leaders to try and get them to maintain wages and production.
Spending money on public works – the issue being that as the federal government increased spending, the state and local governments cut back.
Resorting (reluctantly) to government aid – For example, he created Reconstruction Finance Corporation, which bailed out banks and provided loans to railroad corporations.
Reassuring the public and rousing their spirits – Hoover was acutely aware that people needed to feel confident and hopeful, but he was unsuited the task. He wasn’t charismatic and many of the things he said were odd (such as stating that people in New York had left their jobs and were finding they were making more money as apple vendors) or pessimistic or just wrong (stating that the United States “can’t legislate out of the Depression anymore than we could a hurricane or tornado,” or that “At least no one has starved”).
Hoover was concerned that if the direct federal relief were provided to people, their individualism and self-reliance would dissolve. But people wanted help, and Hoover wasn’t addressing their needs and he grew increasingly unpopular. You might have already heard of Hoovervilles, but people also referred to jack rabbits as Hoover hogs and newspapers as Hoover blankets. The Bonus Army fiasco in the Spring of 1932 basically ruined the rest of his reputation and any more re-electability he would have had.
FDR, on the other hand was charismatic, and very willing to experiment and expand the powers of the federal government. He did work off of some of what Hoover had already done (such as expanding the powers of the Reconstruction Finance Corporation, for one), but he tried more. FDR’s first hundred days is famous because of the amount of laws passed and actions taken (the “alphabet soup” of the NRA, AAA, CCC, TVA, FERA, FDIC, PWA, the Bank Holiday, taking the country off the gold standard, etc.).
And he made gave people hope and confidence. During his first fireside chat on March 12, 1932, he spoke to the American people on the radio and was relaxed, referred to them as “my friends,” and spoke to them of the issues as if they were neighbors—he explained the banking crisis and the planned bank holiday in less than fifteen minutes and urged Americans to put their money back in the banks when they reopened, which they overwhelmingly did. FDR didn't solve the Great Depression by any means, but he provided people relief and a sense of security they hadn't had before.
Sources:
William E. Leuchtenberg, The Perils of Prosperity, 1914-1932 (1993).
Robert S. McElvaine, The Great Depression: America 1929-1941 (1993).
Donald A. Ritchie, Electing FDR: The New Deal Campaign on 1932 (2007).
David M. Kennedy, Freedom From Fear: The American People in Depression and War, 1929-1945 (1999).