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The labour market condition in a region where the unemployment rate skyrocketed from 6 to 10 percent is no better than in the region where the rate fell from 13 to 12 percent.
I can't find a word that fits there except "skyrocketed". "Soared" or "surged" does not work there for some reason. It just does not sound right to my ears.
I am happy with how skyrocketed sounds, but the word is informal. And I am writing an academic paper.
Both "soared" and "surge" work perfectly fine in this context. Perhaps they don't sound right to you because you've never heard them properly used. Some examples:
6 Metrics Behind Zoom Video's Soaring Stock Price
The last time rates soared like this, the stock market plunged double-digits
and in the same story:
Interest rates are surging in the U.S. and around the world, sending shock waves through equity markets
Figurative synonyms such as rocketed, shot up, swelled, and others are also common.
If you don't want to use figurative language -- and there is no reason why you shouldn't, even in an academic paper -- then rose sharply or increased rapidly both work. Example:
In the future, try doing a Google search on a particular word combination such as "rate surge" before assuming it is not idiomatic. You might find that it's actually quite common.
I'm going to suggest that skyrocketing is actually your best option.
The concern seems to be that "skyrocketing" is too informal for use in academic writing. I don't believe this is really true.
It's easy to find examples of scholarly research using this word, in both social sciences and physical sciences, and it seems to be especially common in economics.
It is a much stronger word than surge, or even soar, implying an extremely large increase over a very short period of time. It's most commonly associated with rising prices or rising rates of occurrence.
While many of the things the word is applied to are considered undesirable, I believe the word itself is essentially neutral in tone. While it's most commonly used with undesirables like inflation, prices, and costs, it can be applied to desirable effects such as literacy and employment rates as well.
A good opposite word for a sudden dramatic decrease would be "plummet"
If you're using fell, the contrasting verb (in the past tense) is simply rose:
From rise:
[Merriam-Webster]
7 a : to move upward : ASCEND
7 b : to increase in height, size, volume, or pitch
Also since you are not using a dramatic version of fell (such as plummeted or crashed), despite the fact that the decrease was fairly small, the same neutral language seems to be appropriate for its complement.
However, if you do want something more dramatic, but you don't like any of the words you've mentioned, there seem to be fewer other options that you'd likely be happy with. Perhaps just using an adjective, such as swiftly rose, would be the most formal.
It seems that you have set yourself two contradictory requirements. You want a dramatic, emotive, journalistic term like "skyrocket", or "surge", but also a term suitable for an academic paper, which should not be dramatic or emotive but descriptive or analytic, plain and simple.
So either you choose an unemotive term like "rose", "increased" or you use a journalistic term like "surged", you can't have both.
You could say jumped to indicate a rise over an unusually short time period.
From jump:
[Merriam-Webster]
1 b : to move suddenly or involuntarily
2 a : to move haphazardly or irregularly : shift abruptly
2 b : to undergo a sudden sharp change in value
