What are some effective ways that staff augmentation noticeably scales up your business?

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The way you used the words “scales up” gives me two methods of replying:

• Scaling up capacity to do work.
• This deals with delivering the work that needs to be done.
• It has nothing to do with the way the business operates and scales. Using an example, you need to hire 100 additional people. Trying to engage them directly yourself will take forever. Alternatively, you can use staff augmentation models (dedicated models like http://remote.com or interim skills augmentation provided by several professional services, e.g. www.ascent.io, BJSS, Kainos etc.) to get as much of those heads as you can to manage and deliver the works needed.
• Generally, these models see you retain the ownership of leadership of how projects are managed, prioritised, delivered, built etc. This means you own all responsibility to assign work, manage quality, and deliver required productive outcomes.
• You can, of course, outsource the delivery of projects too to get work done; however, I did not think this was what you were after. In these models, you define the required outcomes and milestones, leaving the how to the professional services providers.
• Ability to drive internal transformations that modernise systems of work that enable an operation to scale while accessing economies of scale.
• This deals with how you look at the engine of the organisation.
• You focus on leveraging experience from an injection of staff augmentation resources to sit within your organisation, understand the engine, and help you internally digitally transform and navigate the risk to achieve a scalable, performing operation that delivers value to all stakeholders.

It would be helpful to understand which one are you after? Capacity or knowledge?