It depends on how much of a site/page you want.
If you just want a section of one page, then what you are doing is probably the simplest way.
If you want a whole page, you can save the page to you local disc and add that to calibre. For a set of pages, create an index html file that includes all of them and add that to calibre.
If you want a whole page or collection of and you are using Firefox, have a look at the plugin
GrabMyBooks. With that, you click a button on the toolbar when viewing a page to add it to a collection and then later save that collection as an epub. I have only used it enough to see that it works. I'm sure I have sen another plugin like this, but couldn't find it the last time I looked.
If you want a whole website or chunk of one, you can use something like
HTTRack to grab it. If the tool you use builds an index file of the pages saved, adding this to calibre should add all the pages as one book. But, you would then need to do a conversion to the format you want.