I think you are conflating a few things. What you have to remember with the gold standard is that the supply (or probably more correct, holdings) of gold determine the money supply when a gold standard is in place, as does the price of gold (the conversion rate). A reduction in the money supply (fear of deflation, leading to bank runs) and a massive withholding of production are generally the two key events cited as causes of the Great Depression. The later being the reason for the massive public works programs of the time were started.
The inflation of the US currency happened much later. Roosevelt banned private holdings of monetary gold and gold certificates, incentivized gold imports and gold production which lead to the inflation. That was nearly 5 years into the depression though.
> supply (or probably more correct, holdings) of gold determine the money supply when a gold standard is in place
It does not when the money is not convertible to gold. Such is a fictitious gold standard.
The bank runs in the Depression were caused by inflation devaluing the value of a dollar by about half since 1914, yet was still convertible to gold. People suddenly realized they could DOUBLE THEIR MONEY by converting their paper money to gold. So they ran to the banks to do this. Of course, there wasn't enough gold to support that, and the banks collapsed. The runs did not stop until FDR made it illegal to exchange dollars for gold.
> The inflation of the US currency happened much later
Check the historical inflation figures. The dollar lost half its value 1914-29.
The inflation of the US currency happened much later. Roosevelt banned private holdings of monetary gold and gold certificates, incentivized gold imports and gold production which lead to the inflation. That was nearly 5 years into the depression though.