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Don't overthink.  Your situation is not that complicated.

Do not start an S-corp without significant legal and tax advice from an expert attorney.  It is a major step, and will provide no tax savings in your situation if you are honest.

The idea with an S-corp is that the hospital contracts with your corporation, of which you are the sole employee and stockholder.  You pay yourself a salary, and if there are extra profits, they are taxed at a reduced rate.  But as a locum tenens, your only product is your own labor.  If a hospital is willing to pay $100/hour, that is your fair market rate, and trying to pay less tax by saying "my fair salary is $50/hr and the other $50/hr is profit to be taxed at a lower rate" is a huge audit red flag.

Forming an LLC may give you some liability protection, but that is also basically irrelevant as a locum.  If you are a sole proprietor as a building contractor and you go bankrupt owing clients money, your personal funds can be seized in bankruptcy.  If you make an LLC, you may get some protection for personal assets if the business fails.  But the amount of protection depends on the state, and the type of business you are doing.   Since you already have, or should have, medical malpractice insurance coverage through the places you work, and you are unlikely to "go out of business" while owing your clients money, forming an LLC is going to be of very limited practical value. 

An LLC with one member files exactly the same tax return as a sole proprietor who has not taken any special steps to form their business.  LLCs may provide some legal liability protection in the state, but the IRS ignores LLCs for income tax purposes.  

In any case, seek legal advice in your state before forming an S-corp or LLC.  I'm sure there are other locums who have accountants help with their taxes, ask around.

As a self-employed person, you can deduct from your income, any expenses that are "ordinary and necessary" for that type of work.  

Business expenses need not be required to be considered ordinary or necessary. Generally, ordinary refers to expenses that are common in the industry and most business owners in the same line of business or trade would normally incur them. Necessary means that the expenses are appropriate and a business owner might not be able to manage without making the expenditure.

Depending on where you work and what kind of work you do, your expenses will almost certainly include mileage (transportation) from one job to another.  Exactly what mileage is deductible is covered in IRS publication 463.

https://www.irs.gov/forms-pubs/about-publication-463

Other ordinary and necessary expenses might be insurance (if you have your own separate policy), scrubs (if you buy your own), a laundry service (if you wash your own scrubs and pay a service instead of washing them at home), medical equipment you use in your work like a stethoscope, blood pressure monitor, otolaryngescope, sanitizing wipes, or anything else you use in providing medical care that is not reimbursed or provided by your client.  (And just for clarity, when you are working as a locum, you are the business, and the medical facility you work for is your client, to whom you are providing services under contract.)  If you have items that are used both in your W-2 job and your locum work, you must allocate the cost proportionally.  (For example, if you work 40 hours a week as an employee and 10 hours per week as a locum, and you use your own stethoscope at both jobs, you can deduct 20% of the cost of a new stethoscope as a business expense.)

The best way to find an accountant to help with these issues is by asking other people in the same situation as you.

You really don't need a CPA (a "certified" public accountant is someone who has an extra certification to do accounting work for publicly traded companies), just an accountant who is familiar with your type of work.  And you likely won't need one very much.  The bookkeeping and taxes are fairly straightforward, once you have an initial meeting to get everything set up so you know what you need to know.