How to charge a client for outsourced work on hourly contracts?

I run a small web development company.
I take on a few clients and do much of the work myself.
Typically I build e-commerce shops or basic wordpress sites.
Sometimes the development is beyond the scope of my expertise, in which case I have the following two options:

1. Spend time learning how to do it myself (longer)
2. Hire a contractor i.e. outsource (shorter)

In the second scenario (hiring a contractor) I typically will hire from an outsourcing site. The hourly rate of a contractor is much cheaper than my own rate. For example I have a maintenance rate of $100/hr, I have found a contractor who can do the work for ~$25/hour.

For maintenance jobs where I bill by the hour, I charge $100/hr (my management rate) + cost of contractor.
E.g. if the contractor spent 10 hours of work, and I spent 1 hour managing the total bill will be: $350.

Although this feels like the most legitimate way to structure my billing, it seems a bit odd that my client will pay $350 for a job that should really cost ~$1000 (or more) if I were to do it myself. After all it took ~10 hours to accomplish the job.

What other billing structure can I implement for the outsourced jobs?

Any insight is greatly appreciated.
Thanks.

First I would make sure your contracts specifically allow for outsourcing the work as you need.

Then, one thing you should understand, is you should only be billing your rates. If you're outsourcing the work, you're really sub-contracting it. The way subcontracting works is your client pays you your rate, you pay the sub-contractor their rate out of the amount they pay you. You don't add it on top of the amount they're paying you. It's unethical. If I had a contractor do this to me I would immediately sever all ties and become very loud on the internet. Subcontracting is an agreed form of business, but since you're not doing the work you don't get your full fee. You're acting as a middle man.

The way I see it is you can either

1) Do what Randy said

2) Charge by the job, or 3) Charge your client a flat hourly rate for everything and take care that the subcontractor will work at or below that rate.

To summarize, as a customer, I really don't care that you subcontracted that portion of work out, I hired your company. As long as you get the job done legally, professionally, and ethically and actually did great work then we are square.