Generally speaking a better route will switch traffic back. I assume you have a default route out of your network via the inside interface of the ASA 5506. It then has a default route out to the Internet via your local ISP.
If you put a more specific route (i.e. something better than 0.0.0.0 0.0.0.0 default route) in your local routing table that will make the traffic prefer that path.
Depending on what the local infrastructure is, where that route is activated varies. If, say you have a core or single layer 3 switch, you put it there.
You can make it switch over semi-automatically using something like an IP SLA monitor that gives you a tracking option that is then use in a backup static route.
A diagram or more complete explanation of your branch office network would allow us to give a more specific recommendation.