Amongst the alternatives for applicants who don't match, MPH is a popular choice. MPH or MHA in itself will not help you at all. Its what you do in those 2 years (publications and more importantly contacts you build) will go a long way to help secure a residency.
|GRE, TOEFL, USMLE, MHA, MPH and residency applicant|
There are MPH programs that accept applications through SOPHAS (Schools of Public Health Application Service . Though individual school requirements may vary, there is at least a few programs like UMDNJ (University of medicine and dentistry of New Jersey)  that allow for application to their school of public health without the GRE . According to the requirements page on their site , in lieu of the GRE - MCAT, GMAT, DAT, or Pass USMLE, Step I; or have U.S. or Canadian Medical Board Certification; or have a doctorate from an accredited school in the US or Canada; LSAT for JD/MPH applicants or the LSAT can be substituted in all departments except BIST and EPID. TOEFL is required for students educated outside the US in Non-English speaking countries, or a passing score in the English Portion of the ECFMG.
One thing to note is that the "English portion of ECFMG" is now an obsolete test and has been superseded by the Step 2 CS which tests demonstration of data-gathering and communication skills (including spoken English). Thus applicants are likely required to take atleast the TOEFL when applying to these programs as Step 2 CS is still not an accepted alternative to TOEFL. After hearing from some other applicants university of texas and few other universities will also waive the GRE requirements of you are ECFMG certified.
Details of the criteria for application to MPH program at UMDNJ are at program website
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