You would end up leaving most of the bright regions of the Milky Way within the so-called 'solar circle'. The vast majority of the bright nebulae would be clustered to one side of the sky. At 10 kiloparsecs, you would be outside the Perseus spiral arm which is the next one out from the Orion Spur where the Sun is located. There is more matter and stars beyond. You do not exit the Milky Way completely until you get about 50,000 light years in the tangent direction.