Hotheaded Giant doesn't check whether you've cast another red spell until Hotheaded Giant enters the battlefield.

Although Hotheaded Giant says it looks for "another red spell," there's no requirement that Hotheaded Giant actually be cast as a spell (or be red) for this part of its ability to work. For example, if a spell such as Zombify puts Hotheaded Giant directly onto the battlefield, its ability will still check whether you've cast a red spell this turn even though you didn't cast Hotheaded Giant itself as a spell.