Erik ten Heg has no shame, I believe that for him to get another job it will take time. This new manager will definitely get that praises because is normal for Manchester United fans and one of their problem is over hyping anything that shows any sign of positivity, they won't observe how it will be like in a long run, you'd see them talking as if they have seen the future and the fans can even say the new manager will win them champions league and EPL title, is really funny. Man UTD can't start doing better immediately the new manager comes in but one thing is for sure that whoever has a good record outside EPL when you start playing or coaching Manchester United you'd fail because the expectation will make you start making mistakes.
You are correct that Erik ten Hag will have a difficult time finding another employment because if they see his performance against Manchester United, no club will want to hire him, except for a little club that is slowly growing and will not compete in the Premier League. But if people look at his success at Ajax, they will take him quickly, but if he goes to Manchester United, I don't think he will get the job as readily. Manchester United's bad performance can be attributed to a lack of a good coach. Since his dismissal, the team has not lost any matches. This weekend's game will be against Leicester and am sure they will still do there best and defeat them again, because they are playing at home and they won't try to lose any point currently, because if they lose and make Leicester defeat them they will drop from 13 position to 14 again, and probably if west ham win he will be a poor position for Manchester United. so am very sure they Manchester United will do there best to get three point this weekend.
I would assume that for the next job he will probably not have that much trouble. You do not go from United to a small club directly that easy, it takes a while for people to get there, I believe that we are going to end up seeing him be at a decent team first, not as high as United, but not that terrible of a team.
The real crash happens when he is hired by another decent team and ends up with a terrible result there as well. So, after this United job, and another being bad? Then he will be some Saudi league level manager and will not be good at all, he will not be hired by any teams, even relegation zone teams will scoff at him and will not expect him to be good. But if he is hired by some decent team, and does wonderful there, then he can recover his career and convince employers that it wasn't him that was the problem, it was the united management or even united players and not him, and find another good job, all depending on how he does on his next job.