How can I brighten a backlit subject’s face while keeping rim light in the hair?

Yes, this look can often be done without flash if you shoot at the right time. The easiest approach is to photograph near sunset, when the sun is low and weaker relative to the open sky. That lets you keep the warm backlit glow on the hair while the face is filled by softer, cooler skylight.

In the example you posted, the face light looks soft and slightly cool, which suggests open sky fill rather than direct flash. A reflector is also possible in some similar shots, especially if the face light looks warmer and harder.

So the best technique is usually:

• place the sun low and behind the subject for rim light
• shoot close to sunset
• use the open sky in front of the subject as natural fill
• expose carefully for the face while keeping the backlight attractive

Flash is not necessarily required here. If you do use flash, softer light generally means making the light source effectively larger or less direct, but the provided answers suggest timing and natural skylight are the simplest solution for this specific look.