Employers are obligated to pay full-time employees health insurance on day one throughout their employment. Before I started at my job I had to complete a drug and blood work screening to confirm my eligibility for insurance. The coverage started when the results came back. I've never heard of having to work for a total of 90 work hours, especially with a government job.
Perhaps you're eligible to buy into their insurance plan after you've worked for 90 work days because you're not full-time? Obama care is such garbage legislation. This is why there's so many 32, 36, 38 hour/week jobs, all so employers can skip out of paying for insurance because anything below 40 hours/week isn't full-time.
Obama care was supposed to expand medicaid eligibility, yet in my state they removed half of the qualifications for medicaid because of Obama care. Accordingly, most hospitals offer something called "free-care" mainly for those who can't qualify for medicaid. The % of coverage from a hospital's free-care is depends on income... but if you're really desperate to start HRT, ask a psychologist if they're open to doing an eval (an eval is only a visit or two, not having to pay for visits every week to talk about what you already know) to confirm you're actually transgender and sending a letter of recommendation to an endocrinologist. Call an endocrinologist's office at the hospital you have free-care at and make an appointment. I'm unsure if prescriptions are covered under free-care plans, but hormones aren't all as expensive as people think unless transition isn't your top priority.