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I am an American citizen and my husband is a British citizen from Scotland. I am applying for a spouse visa int he UK in order to live with him but am worried it might be rejected due to the financial requirements. He cannot provide proof of making over £18,600 last year, but can provide proof of just starting a new job that will make well over that this year. Will that proof suffice, or is the requirement solely based on the previous year? 

If your spouse has just started the job, then it really comes down to what you mean by that.

From the UK Government advice on Family Visas:

What counts as income

You and your partner can use:

• income from employment before tax and National Insurance (check your P60 or payslips) - you can only use your own income if you earn it in the UK


• ...

However, this is qualified with a duration:

If you’re using income from self-employment or employment, you’ll need to prove you or your partner received that income for 6 months or more.

Example
You’ve worked with the same employer earning £18,600 or more for 6 months or longer.

So it looks like future earnings are not taken into account unless the job is stable.

There are other caveats on that page which may or may not apply to yourself, so I would suggest reading it and the detailed guidance.

Part of the detailed guidance does allow for this scenario, if the salaried or non-salaried income from your spouses current job plus the salaried income or non-salaried income from your spouses previous jobs in the last 12 months exceeds the £18,600 limit.

5.3. Category B: Less than 6 months with current employer or
variable income – person residing in the UK (page 24)