I just challenged an appraisal this month. Appraised value increased from $164k to $229k. Here's how I did it in 6 steps:
1). Review the appraisal report, look for better comparable properties, inaccuracies in the report, and overlooked upgrades in the subject property. Carefully review the comparable properties used by the appraiser. In my case the appraiser failed to use the home home next door that sold the month before and instead used homes that sold a half mile away. Additionally from reading the original descriptions from the realtor MLS listings I found that one of the comparables was sold at auction.
Look for anything inaccurate such as the sqft or the appraisers assessment of the condition of the home, perhaps there are upgrades that need to be be brought to the appraisers attentions.
2). I submitted the better and closer comparables to the appraiser along with a list of renovations and upgrades that were overlooked. The appraisers response was to disregard these findings. (Sometimes step 2 has a better outcome).
3). Confident the original appraisal was wrong, I worked with the mortgage broker and a new appraisal was ordered at my expense.
4) New appraisal came back 35% higher with the use of more accurate comparables. Mortgage broker has requested a refund from the first appraiser due to their incompetence.
5) Await completion of cash out loan at corrected higher appraisal. Await refund check for the original appraisal.
6). Reinvest the proceeds and assure your spouse that you are not going out of your way to add more stress to your life, just the opposite. :)
Good luck!