As with almost anything, there are people who care and others who don't! But it is definitely a thing to consider if you are trying to write well.
Degrees of severity
There are two angles to this. One is “trivial“, in that the consideration is mostly about legibility, flow, and aesthetics. The other is more consequential, where the “false compound” could really throw off the reader, complicating the interpretation process, and in the worse case cause them to misinterpret the sentence.
In your example, 「去年教職」 is unlikely to cause a serious problem, at least for native readers. 去年 is a common enough word, and 去年教職 is hard to conceptualize as a compound (what would that even mean? I guess it could be read as 「去年教/職」 like a clergy job?). On the legibility side, it helps that 去年 and 教職 are demarcated by a difference in 画数-density. Still, I understand why one would want to break up the false compound.
False 3-kanji compounds tend to cause more confusion:
最悪値を変える (At worst, I'll change the value?) (I'll change the worst value?)
あなたとは当面面と向かって話せない (I won't be able to talk to you in person for the time being)
The “proper” quick fix is to insert a comma:
This is good because it matches the flow of how you would read it out loud.
Where the medium permits, you can “cheat” and insert a narrow space:
This is done in informal writing like hand-written letters and online communication. It's generally not allowed in the body text of publications (header text can be an exception).
Some words read better when switched to hiragana:
到底底力は出せない → とうてい底力は出せない
Choosing between kanji/hiragana is an art in itself and a serious writer can be a real stickler for these things. Prestigious publishers may have house rules. But it's a great technique to build.
Opt for a synonym:
最悪値を変える → いざとなれば値を変える
Reorder the words:
Now we're into rewrite territory, and this will start to affect the meaning of the sentence.