You can certainly request an insurance contract that is valid from July 1st, or from the day you arrived, even though these dates are already past. My health insurance was made valid retroactively from the date I entered the country even though it took a few months after I arrived to sort everything out. I just ended up owing a few months backs payments for premiums for these months.
I your case, I'd suggest contacting your insurer to request they send you a contract for insurance beginning July 1st , then visit the doctor. Save all your receipts from your trip to the GP and for any tests, treatments or meds you require and send them into your insurer once you've signed your contract and paid your monthly premiums. Your insurer will then figure out who owes what to whom, (depending on what your insurance excess is).
Have your checked out the
Healthcare FAQ? It explains the Swiss system, and also has links to documents that will tell you what services will be covered by your basic insurance and what won't.
Cheers,
HeatherM