Two things that are true about Smoot-Hawley:
(1) It wasn't a good idea.
(2) It didn't cause the Great Depression and probably didn't even deepen it that much.
See my posts at
https://www.alternatehistory.com/forum/threads/no-1930-smoot-hawley-tariff-act-no-great-depression.324772/#post-9529921 and
https://www.alternatehistory.com/forum/threads/ahc-wi-hoover-as-a-multi-term-president.348959/#post-10976875
But in any event that is irrelevant to the differences between a Coolidge administration and a Hoover one because Coolidge was a protectionist and would undoubtedly have signed the bill.
I'll recycle two old posts of mine:
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(1) Coolidge's whole record was pro-tariff; he not only supported Fordney-McCumber
https://en.wikipedia.org/wiki/Fordney–McCumber_Tariff but implemented it in a very protectionist way: "Fordney-McCumber let the president raise or lower individual tariffs, and when Coolidge used this power he almost always raised them. Coolidge also inherited (and declined to change) a Tariff Commission populated with representatives of the industries it controlled—-an unholy arrangement that lasted until eventually Congress cried foul."
https://books.google.com/books?id=ogc9EZf8Ry8C&pg=PA73
***
(2) AFAIK Coolidge never expressed opposition to Smoot-Hawley. Even a conservative defender of Coolidge states "We don't know his stand om the Smoot-Hawley tariffs."
https://books.google.com/books?id=4CKHfGfNhIQC&pg=PT220 As I note below, Coolidge was still defending protectionism in late 1932, a few months before his death.
Nor would there be any reason to think, even based entirely on his record through 1924, that he was a free trader. Where did you get the idea that he vetoed a tariff bill? His only vetoes in 1924 listed at
https://www.senate.gov/reference/Legislation/Vetoes/Presidents/CoolidgeC.pdf were of spending bills, notably the bonus bill. He wasn't altogether happy about the Revenue Act of 1924, but that was because he thought it didn't cut the income tax enough (and anyway he did sign it)--it had nothing to do with tariffs. Coolidge was worried that the government was taking in too much revenue, and he wanted to stop that by lowering income taxes, not lowering the tariff.
In addition, Coolidge's whole record had been protectionist. In 1922, he praised Alexander Hamilton and said that the Republicans were the descendants of the Federalist and Whig Parties, not of the laissez-faire Jeffersonians: "The party now in power in this country, through its present declaration of principles, through the traditions which inherited from its predecessors, the Federalists and the Whigs, through their achievements and through its own, is representative of those policies which were adopted under the lead of Alexander Hamilton
[1]"
https://www.coolidgefoundation.org/...sts-calvin-coolidge-and-the-full-dinner-pail/ Not surprisingly, he praised McKinley's protectionism, stating in 1923 "He [McKinley] at once revised the tariff and strengthened the law establishing the gold standard. Prosperity immediately returned. There was not only a domestic market but immense exports. The foreign trade increased more under the first term of McKinley than it had ever increased in any other four years." " Coolidge praised McKinley's “...application of his principle of a protective tariff, which furnished the initial opportunity for laying down of the greatest industries of America and the development of her entire resources.” Ibid. The GOP platform on which Coolidge ran in 1924 was strongly protectionist: "We reaffirm our belief in the protective tariff to extend needed protection to our productive industries....The enormous value of the protective principle has once more been demonstrated by the emergency tariff act of 1921 and the tariff act of 1922."
https://www.presidency.ucsb.edu/documents/republican-party-platform-1924.
Calvin Coolidge was a protectionist before he became president, as his tributes (while still vice-president) to Hamilton and McKinley showed. He was of course a protectionist while he was president. And he was still a protectionist after leaving the presidency. "The greatest asset of our whole economic system is its effect upon commerce, agriculture, industry, the wage earner, and the farmer, and practically all our producers and distributors, is our incomparable home market. It has always been a fundamental principle of the Republican Party that this market should be reserved in the first instance for the consumption of our domestic products…Our only defense against the cheap production, low wages and low standard of living which exist abroad, and our only method of maintaining our own standards, is through a protective tariff. We need protection as a national policy, to be applied wherever it is required.
[7]" Calvin Coolidge, “The Republican Case, The Saturday Evening Post, September 10, 1932,”
https://www.coolidgefoundation.org/...sts-calvin-coolidge-and-the-full-dinner-pail/ This was just a few months before his death.
Calvin Coolidge was protectionist, protectionist, protectionist. There was no time he could reasonably have been seen as anything else. Period.