Generate uniformly random float which can return all possible values

Well, standard way, I believe, is to generate up to 1074bits integer and map it to the double. Beware, that your RNG should have internal state at least 1074bits long.

Reference implementation: http://xoshiro.di.unimi.it/random_real.c

Discussion about it: http://xoshiro.di.unimi.it/

Another good link: https://lemire.me/blog/2017/02/28/how-many-floating-point-numbers-are-in-the-interval-01/

Porting this code (suggested in Severin's answer) is a possible option.

I think that it is equivalent to first generate the significand bits (by generating a random float in [1,2)), and then choose the exponent from a geometric distribution (it has a 0.5 chance of being -1, 0.25 of being -2, etc.).

// uniform returns a uniformly random float in [0,1).
func uniform() float64 {
  sig := rand.Uint64() % (1 << 52)
  return (1 + float64(i)/(1<<52)) / math.Pow(2, geometric())
}

// geometric returns a number picked from a geometric
// distribution of parameter 0.5.
func geometric() float64 {
  b := 1
  for rand.Uint64()%2 == 0 {
     b++
  }
  return b
}

We can probably make geometric() faster by using one of the LeadingZeros* functions from the bits package instead of doing one coin flip per bit.