Is there another place after the very beginning that you can edit your character's face?

Open the in-game console and type "showracemenu" and hit enter.

Be careful if you use showrace menu. Do not change your race, not even checking out another race and then going back to Nord. The moment you switch race, your stats will be reverted.

Using "showracemenu" is safe only if you stick to your starting class and edit his appearance.

I use 360...

This was true in Oblivion. It hasn't happened to me in Skyrim, even when changing race and six. It seems to have been fixed.

wait... so the console command is safe?

i just played OB recently and i thought showracemenu reset all skills even with the class etc left untouched...

Tthis "showracemenu" is PC only, isn't it?

All my stats and perks were intact after a six change, so yeah, I'd say it's fixed. Unless it changed the actual stats, but forgot to change the skill display....

If you use 360 then no.