It's the way things are headed in the future. Increased labour "flexibility" (or disposability). It's the natural culmination of 150 years of capitalism, division of labour and wage slavery. Why hire permanent employees when you can hire temporary workers for less pay and offer no no pension or other benefits? And with a saturated labour supply, local workers are forced to take up ever lower and lower pay. If you're a temp and go from contract to contract, you'll find the pay you're offered for each new contract going down and down. Even retail service and hospitality jobs are now being done through temporary agencies. The agency serves as a "proxy" employer in order to enable the direct employer or client to bypass existing employment laws which were put in place to protect workers. Of course this is all selectively applied to the lower level jobs taken up by the proletariat. The bourgeois professional class -- police, teachers, civil servants, lawyers, etc -- continue to have permanent jobs. And before anyone brings up self-employed people (e.g. IT specialists) who work contracts for very high pay, they are not even remotely in the same category as the low level "temp".
However, to answer your question, some of the more reputable agencies, if they are familiar with you, will be helpful and let you know which company it is if you ask them. There are only a couple of agencies that I have any respect for; the others are not worth the time of day.
I am surprised also that in this day and age companies are quite happy to pay what the emploment agencies charge. I was put forward for my present job by an agency, as I had been temping with them for a year after having been made redundant.
On the final invoice, along with their normal fees, there was a charge of 20% of my starting annual salary!