API-650 doesn't cover cone bottom tanks. The rules in API-620 do cover that configuration. API-620 is intended for low-pressure tanks, but the rules will work on atmospheric tanks as well. You don't necessarily have to do all the API-620 inspections to use API-620 for the design.
Using API-620, it does take some understanding of how the methods work, and it is not just a cookbook approach. In particular, you have to understand how the free-body diagrams need to be set up for a particular configuration, and may need to add terms that are not present in the published equations. A skirt-supported tank designed by API-620 is fairly straightforward.
API-620 does not cover leg design, and I've not seen anything that covers the subject well. AWWA D100 includes elevated tanks of any configuration, but doesn't give you any helpful design information- and does give some misleading design information. Gaylord & Gaylord have a "Structural Engineering Handbook" that shows some hopper design with legs, but the method shown fails to address some major issues (torsion, for example). I've not really seen anything any better, either.
Another configuration you might want to investigate is a scalloped-bottom configuration. However, the detailed design of those becomes very awkward, and approximate designs also tend to leave major items unaddressed.
