I haven't run into this before so I'm trying to figure out reach as a long creature increases size multiple times. The Goliath Druid is able to cast enlarge person on it's animal companion. Assuming a large Elasmotherium animal companion, it should start at 5ft reach. Enlarge Person makes it huge. Does it now have 10ft or 5ft reach? To add more confusion to the matter. In trying to find an answer to this question I came across the Elasmotherium creature entry on d20pfsrd which is huge and has 15ft reach.
https://www.d20pfsrd.com/bestiary/monster-listings/animals/megafauna/megafa una-deinotherium/megafauna-elasmotherium/
What's more, I plan to prestige into Mammoth Rider which nets a possible Gargantuan Elasmotherium with Enlarge Person. How do I handle it's reach?
I believe all Druids can enlarge their companion, as long as they have the share spells ability.
Of course Goliath Druids can still do it even if they archetype the ability out.
Regarding reach, Enlarge Person only grants reach when it makes you Large, apparently. Small to Medium or Large to Huge, etc : no reach.
So, not your rhino either. Good news for people who fight Giants, though.
Mammoth Rider has its own rules, and will increase your reach to 10 ft first (so no effect if it was already that or more somehow), then 15 later on.
Beyond that ...
You specifically can't grab reach with Evolved Companion, but there might other options I'm not thinking of.
Lunge is always a possibility.
Edit : I forgot : what the regular monster versions of animal companions are has no bearings to speak of. Which is why bears were a bit of a sore spot until recently or why elephants companions really aren't that big. Same thing here.
Companions follow their own rules.
As Nyerkh said, enlarge person only says anything about reach in the case that it makes you Large (and that you're a humanoid). But I'd ask the GM to consider using the natural/typical reach column for a Long creature (of whatever size) on the Creature Size and Scale table under Big and Little Creatures in Combat (which works out to 5' reach at Large plus 5' per category thereafter).
That is exactly the table I needed Fuzzy-Wuzzy. Thank you!
Nyerkh, most druids do not have the ability to use enlarge person without multiclassing. Goliath adds it to the spell list and can sacrifice slots to cast it spontaneously. Adds a nice little boost until 5th level spells come around with animal growth.
EDIT: The important part of Share Spells is "Spells cast in this way must come from a class that grants an animal companion." I think there are some archetypes from the casters of enlarge person that can grant animal companions, but usually enlarge person can never be applied to ACs.