Tom suggested me look for another job.
Tom suggested I look for another job.
I’m guessing that the first sentence is not correct. Can I use me / her / his in this kind of subjunctive structure?
For this type of questions, I suggest you use a good dictionary.
For example, if you take a look in the Oxford Advanced Learner’s Dictionary, you will see all the possible structures for this verb.
As you can see from the information provided, you can suggest something (to somebody) or you can suggest (that) + clause but
You cannot ‘suggest somebody something’: [INCORRECT] Can you suggest me a good dictionary?
suggest somebody actually means a different thing:
to tell somebody about a suitable person, thing, method, etc. for a particular job or purpose
So you can say
Tom suggested me for the job.
that means that Tom proposed you for the job. It doesn’t mean he was sharing his ideas with you.
With that said, you need to use a that-clause in your sentence (that can be ommitted, though):
Tom suggested (that) I look for another job.
So you should use I because it serves as the subject of the clause.