The short answer is there is no easy way to get to the jail trail from the Birmingham Bridge.
If you really must though, you can take the ramp down to Forbs avenue/ Oakland. At the bottom go left using the sidewalk to salmon a few hundred feet up to Brady street. Turn left and follow down toward 2nd avenue. There is a gravel/dirt lot at the bottom of the street with a path that will take you up to the jail trail.
Just be careful at the big intersection at the bottom of the Birmingham bridge ramp – drivers are confused on where they are going and they don’t expect to see peds or bikes.