Yes, Alexander Hamilton and the Federalists did generally support the idea of a loose interpretation or construction of the Constitution. They differed from the Democratic-Republicans, led by Thomas Jefferson, who wanted to interpret the Constitution strictly. The Federalists supported loose construction because they were in favor of having a stronger central government.
In the early United States, the two main political parties disagreed about how to determine if the national government had the right to do something. The Democratic-Republicans wanted to interpret the Constitution very strictly. They wanted to say that the government only had the right to do something if the Constitution explicitly said it had that right. This would have limited the powers of the federal government severely. By contrast, the Federalists wanted a stronger central government that could do more things. Therefore, they wanted to interpret the Constitution loosely. They wanted to say that the central government could do things unless the Constitution said it could not.
For example, in 1791, there was an argument over whether the US government could set up a national bank. The Constitution does not say that the government can do this, but it also does not forbid the government from setting up a bank. The Federalists, led by Hamilton, wanted to set up such a bank because they thought it would help the national economy. The Democratic-Republicans thought this would be unconstitutional. The Federalists won and the bank was created. From this, we can see that Hamilton did indeed support a loose interpretation of the Constitution.