I would have it follow under the animal companions share spell ability.
Spoiler:
Share Spells (Ex)
At the druid’s option, she may have any spell (but not any spell-like ability) she casts upon herself also affect her animal companion. The animal companion must be within 5 feet of her at the time of casting to receive the benefit. If the spell or effect has a duration other than instantaneous, it stops affecting the animal companion if the companion moves farther than 5 feet away and will not affect the animal again, even if it returns to the druid before the duration expires.
Share Spells (Ex)
At the druid’s option, she may have any spell (but not any spell-like ability) she casts upon herself also affect her animal companion. The animal companion must be within 5 feet of her at the time of casting to receive the benefit. If the spell or effect has a duration other than instantaneous, it stops affecting the animal companion if the companion moves farther than 5 feet away and will not affect the animal again, even if it returns to the druid before the duration expires.
Additionally, the druid may cast a spell with a target of "You" on her animal companion (as a touch range spell) instead of on herself. A druid and her animal companion can share spells even if the spells normally do not affect creatures of the companion’s type (animal).
By the rules, the animal companion doesnt get the perk. Most animal companions have speeds of 40+, so even at half, the keep up with the druid wearing hide armor.
Now, an easy fix that can be applied is to change the woodland stride ability into a spell, then it can be shared. I beleive a spell version exists in the Spell Compendium, for rangers or such, but that might do the job.