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.

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