These two words are often confused. Rein is related to horses. Reign is related to kings and queens.
So where does the confusion lie?
In a phrase like reining in – he reined in his activities because he was exhausted – reining in refers to holding the horse back, restricting its movement.
In a phrase like free rein – he was given free rein to organise the office as he wished – free rein refers to slackening off the horse’s rein so that it can move more freely.
But confusion can arise when you consider that being given free reign could also refer to being free as the ruler to do what you want.
So consider whether you are talking about rein as in a horse-related metaphor or as in a king-related metaphor. There are places where either would be suitable, but there are definitely places when only the horse one makes sense!