Temp firms lose business when economy weakens.
They gain business when economy begins to recover.
They lose business when economy stabilises.

They have started gaining business.

Conclusion: Economy must be recovering.

By concluding that the economy must be recovering, we are assuming that they gain business only when economy recovers. That there is no other condition that could lead them to gain business.

A) Temporary-services firms are more likely to regain old clients than to acquire new ones when the economy begins to recover

Old or new clients is irrelevant. We are talking about business in general.

B) Temporary-services firms do not gain business when an already weak economy worsens

Correct. We are assuming that temp firms do not gain business in any other condition. So we are assuming that they do not gain business when an already weak economy worsens (another condition).

C) New companies do not often hire temporary help until they have been in business for some time.

Who gives business to these temp firms is irrelevant.

D) Companies that use workers from temporary-services firms seldom hire those workers to fill permanent positions

This option tells us that companies hire these workers for temporary positions, not permanent ones. So when their business needs change the positions can be removed/added back etc. Do we need to assume this? No. We don't care which positions are filled by these temp workers. Our concern is the number of positions companies have for these temp workers.
Even if companies use temp workers to fill some permanent positions and keep switching people on these positions every few months/years, it doesn't matter. We already know that the business of temp firms changes as per the status of the economy. Why and how that happens is outside our scope (temp positions are removed when business goes down, new positions are added when business starts recovering etc).
We need to focus on only the gap in our reasoning. Whether temp companies gaining business means economy is recovering.

E) Temporary-services firms can most easily find qualified new workers when the economy is at its weakest.

Irrelevant how temp firms hire workers.

Answer (B)