QZ 8501 took off from Surabaya for Singapore at 05.35 local time. That ment the crew almost certainly reported for duty at 04.30. They had a double so they were scheduled for an eleven hour day. Their natural body clock would not have permitted them to get eight hours sleep before getting up before 04.00hrs.
Once airborne they were now under pressure to get to Singapore and back, twice, without overrunning their duty hours. When approaching a massive build up their first reaction would have been to go around it but then the time penalty would start to register. The desire to stay inside their time bracket would encourage a 'suck it and see' approach and maybe they tried this.
When it became obvious that this was not going to work that is when they asked for a heading and climb.
It was too late.