It was the insurance company, Olympus Mons Insurance out of Marsport, who demanded Aran Jones’s presence aboard the Outcaster. The exploration company Olympus Mons was insuring had lost one ship already and narrowly missed losing another. Aran, an experienced captain and, more importantly, a non-Mars Born one, was to see that the Outcaster fulfilled its mission charter safely and returned.
“You’re not prejudiced, are you?” the company official asked.
“The war’s been over for fifty years. Most of the crew on the Outcaster weren’t even born when their parents were shooting at me and my buddies,” Aran said. He was well aware of the fact that there would be another less overt, religious, chain of authority aboard the ship. Sol System Explorations Ltd. was

I think I remember an earlier version of this from waaay back, lol, though I can't remember the specifics for direct comparison. Anyway, here are my thoughts:

'... aboard the Outcaster. The exploration company Olympus Mons was insuring had lost...' - at the very least, stick a 'that' before 'Olympus Mons' for clarification, but I'd suggest shortening the whole clause, somehow. Indeed, why not simply cut it to 'Sol System Explorations Ltd had lost one ship...'? You'd have to change the bit further down, of course, but that's not too difficult to overcome ('He was well aware of the fact that there would be another less overt, religious, chain of authority aboard the ship, and was being well paid to ignore it' / '... aboard the ship. He was being well paid to ignore that fact', etc). Anyway, as it is, it's a bit hard to follow.

'Aran, an experienced captain and, more importantly, a non-Mars Born one, was to see that the Outcaster fulfilled its mission charter safely and returned' - slight infodump here. That aside, it's a largely redundant statement - of course the captain is there to see that the ship completes its mission and returns home safely; that's what captains do. Even if he's just overseeing the mission (not actually captaining the vessel), his rank implies that he'll be there to keep things running smoothly. With that in mind, the only important things here are 'experienced captain' and 'non-Mars Born', so find a way to keep those and discard the rest.

'another less overt, religious, chain of authority aboard the ship' - not at all sure about the way you've inserted 'religious' into the sentence, there. I think it'd be ok to have it in brackets or between hypens ['less overt (religious) chain' or 'less overt - religious - chain'], but I don't think you can use commas for parenthesis like that - parenthetic commas (if you like) are for subordinate clauses, mostly. I could be wrong, of course, but I think that's how it works.

***

That's all that springs to mind - it gets my interest about half way through, but I think your very first lines could do with some tightening.

Hope this helps .

But the first two sentences are difficult to read.

Since Olympus Mons has the word "Insurance" in their name, you can likely dispatch with calling them an insurance company. And perhaps be more direct. Maybe something like: Olympus Mons Insurance out of Marsport demanded Aran Jones’s presence aboard the Outcaster

(should it be Jones's or Jones'? I'm not sure)

Then in the 2nd sentence sentence, you could just refer to the previously mentioned insurance company as "them". Maybe: The exploration company they were insuring had lost one ship already and narrowly missed losing another.

Should non-Mars-Born all be hyphenated?