So and that are independent Complementizers -- words or phrases that identify (and often delimit) various types of subordinate clauses -- with separate syntactic trajectories. They happen to come together in this construction, but there is a lot of variation.
That is the same that that (optionally) appears introducing the tensed Object Complement clause the Earth is a rhomboid cylinder in
- She believes (that) the Earth is a rhomboid cylinder.
and that used to appear before other kinds of tensed clauses as well
- Whan that Aprille with his shoures soote / The droghte of Marche hath perced to the roote
Nowadays we can't say
- *Tell him when that you will see him again.
but we can say
- Tell him so that you will see him again.
So is a frequent and quite general complementizer, a "pro-adverb", if you will, marking pretty much any clause or reduced clause (i.e, phrase) with implied causation, intention, or responsibility -- human, natural, logical, accidental, or conversational -- such as might be expressed more precisely with adverbial clause conjunctions like
because, anyway, therefore, as I was saying, hence, now, regrettably, surprisingly, thus, ...
So that can therefore introduce any such tensed clause and thus means the same as in order that (unlike so, in order can appear also with the infinitive complementizer to, as in order to).
Order has a variety of deontic modal meanings -- orders are imperatives, and Imperative is a Mood, after all -- and therefore the OP's intuition that modal verbs are involved is correct. Like many constructions in English and other languages, there are often presupposed modal meanings that show up in lexical choices.
For instance, English relative infinitives almost always imply some kind of weak deontic modality, like the modal auxiliary should or ought to:
- Bill is the man to see about that. = Bill is the man whom one ought to see about that.
- Bill is the man to do that. = Bill is the man who should do that.
Since so is so non-specific, the details of precisely what kind of purpose, result, causative, or summatory clause relationship obtains is left, like so much else, to the listener or reader to guess at, from what they believe about the context.