Using a courier service gives you many different options, especially if you have a multi-stop shipment. You can have the courier fit your deliveries into their most efficient route, for one. This means that the courier will stop and make other deliveries while delivering your packages. If you don't like that idea, you can pay for a dedicated courier. This will cost you a bit more, but it means the courier will work only for you that day, delivering your packages to your locations as necessary. With many dedicated couriers, you can even specify the route the courier takes to deliver your packages, placing the most important at the top of the list.
Your Miami courier service may also offer shipments outside of Miami or even Florida. If it's a fairly large company, it may even be able to ship your packages anywhere in the country. However, sometimes you may actually find a better option in another state. Is it possible to ship something from one courier company and then switch over to another in mid-shipment?
Well, yes, you can always have a courier service deliver to several of your destinations in a multi-stop shipment and then have the remainder of the shipment be delivered to a terminal that is leased by multiple shipping companies. One of these other companies could then take on the shipment. Sometimes this happens anyway; a large shipping company may only deliver your packages to a large city. A smaller courier company may then take those packages to smaller cities or other nearby locations that the larger company does not service.
However, all in all, it's not very efficient to do this. Changing your packages from one truck to another takes time, as does the paperwork to transfer packages from one service to another. Plus, unless there's a good reason for it (as in transferring packages from a large service to a smaller one for delivery), there's really no advantage to it. It may not always be possible, either, especially if the courier service only uses dedicated terminals that they completely own or exclusively lease.
Then there's the issue of tracking your packages; you would need multiple reference numbers since courier services do not use the same internal tracking numbers. This would make it nearly impossible to track your packages online and would make it very difficult to track them down if they were to go missing. Overall, it's much easier to stick with one courier service for your multi-stop deliveries.