As someone already mentioned, your betta friend may just be shy because he is adjusting to his new home. Moving is stressful for all creatures... even fish.
However, in my experience bettas are generally very brave and curious little critters, and the prospect of exploring a new environment is generally pursued with relish. The first thing I would suspect is an uncycled tank or an illness in the fish.
Test your water parameters and inspect your betta for any signs of illness (discoloured markings, pale colour or a brown or green hue that was not present before, bumps, frayed fins, etc.). Also be sure that the temperature is somewhere between 78 and 83 degrees F (they like it best at about 80). Photos would be helpful to those of us replying here if you see anything out of the ordinary.
One thing I always provide in a new tank is a small coloured cup or bowl, turned on its side and lined in the bottom with smooth flat rocks, where my new fish can go and hide and feel secure. Don't use clear glass, and face the opening away from the water current so that it is still and serene inside. I usually try to place some plants near the opening to "veil" it and contribute added feeling of security and privacy.
Hope this helps.
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