When restoring a character's hit point maximum lost from exhaustion, does the character also regain hit points?

For example, the fourth level of exhaustion reduces a character's hit point maximum to half. Normally, a long rest will remove one level of exhaustion and conveniently allows a character to regain hit points. However, what happens when a character uses a spell like greater restoration to undo the effect? Does the character regain half their hit points or do they stay at half hit points?

I took a quick look through the PHB but cannot find anything on the effects of regaining maximum hit points. Wording for spells like aid imply hit point maximums and hit points are entirely separate.

Current hit points are unaffected by hit point maximum increases

A character's hit point maximum is only the upper limit on the creature's current hit points:

A creature's current hit points (usually just called hit points) can be any number from the creature's hit point maximum down to 0.

This is reinforced in the section on healing:

When a creature receives healing of any kind, hit points regained are added to its current hit points. A creature's hit points can't exceed its hit point maximum, so any hit points regained in excess of this number are lost. For example, a druid grants a ranger 8 hit points of healing. If the ranger has 14 current hit points and has a hit point maximum of 20, the ranger regains 6 hit points from the druid, not 8.

Things that increase your hit point maximum (or remove an effect that was reducing your hit point maximum) have no effect on your current hit points unless stated otherwise.

This is reinforced by the wording of the aid spell:

Each target's hit point maximum and current hit points increase by 5 for the duration.

Note that aid separately increases your max HP and your current HP. If increasing your max HP automatically increased your current HP as well, it would be redundant and potentially confusing to state both parts separately. However, because your max HP is just an upper limit, and increasing it doesn't increase current HP automatically, it's necessary to state in the spell description that both are increased.

In short: max HP increases don't change your current HP unless stated otherwise (as aid's description does).

(Note that hit point maximum decreases can decrease your current hit points, because your current HP can never be above your max HP.)

You don't automatically gain HP

Crawford answered a question on Twitter regarding what happens when the max-HP-reducing effect of Life Drain wears off:

When Life Drain wears off, the subject's hp maximum is no longer reduced, but there is no automatic restoration of hp.

This supports the fact that when an effect that reduces your maximum HP is removed, you don't automatically gain HP. Since exhaustion and life drain reduce maximum hit points in the same manner, this ruling also works for exhaustion.