A .38 is just another version of a 9 mm which is the baby brother of the .357 magnum. The cartridge is more than adequate to render the brain mush especially with the use of a hollow point. But.... here is my concern .... I have read reports of policemen being shot in the ear with a 38 and the bullet being redirected by the mastoid bone into the jaw. Moderate caliber firearm suicide is usually most effective when the brain stem is actually hit - so that is why "intraoral" shots tend to be favored - less bone between the gun and the brainstem. However, the odds of survival with a .38 above the ear, assuming no bone deflection, and assuming proper trajectory, is going to be very low. Wheelsonthebus has done an analysis and his information will be more indepth than what I am presenting.
But, and forgive me if I am overstepping my bounds on a suicide site, I am not a fan of guns for suicide because they are the most readily available CTB method for impulsive actions. I believe suicide should never be impulsive; it should be the last resort and only after careful consideration. Even though I have access to many guns, I have committed to others I care about to never use them for CTB. They are aware of my mental health struggles but they are also aware that I try not to break my word. Suicide is a one and done act - and really all death is just a question of timing. I am a fan of the saying - Life is a sexually transmitted disease that is 100% fatal. So, I urge people not to be impulsive - we are all living on borrowed time anyway.
Edited to clarify that a .357 mag can fire a .38 round. They are much cheaper if you do a lot of target shooting.
Edit #2 - but....you can't fire a .357 out of a .38 pistol so don't try it - even though I don't think you could get it to fit, but assuming you managed to jam it in, it would probably blow the gun up and your fingers with it.