Were they doing that after the SR1 was destroy or was that after Shepard dealt with the collectors?
That depends on if the player has the Arrival DLC and when they do that DLC (which can be done in ME2 long before Shepard deals with the Collectors)... because, according to Liara, she and Hackett realized they were running out of time after Shepard had to destroy the Alpha relay.
Quote Liara speaking to Shepard on Mars when he inquires about why they were looking in the Mars archives: "Process of elimination, mixed with a little desperation. When you destroyed the Alpha relay, you bought us some time, but then you were under investigation. I knew I had to do something, Hackett knew it too. He contacted me asking if I would use my resources as the Shadow Broker to find a way to stop the Reapers..."
Also, a "process of elimination" implies that they searched several different sites before searching on Mars. Also, the Shadow Broker's resources (i.e. agents) are portrayed as being quite large in number and coming from a variety of species, so it can't really be said that only humans were doing anything either, but it is clear that it was Hackett (not the council or the Alliance) who was personally responsible for recruiting Liara and her team to that task. There is still room, however, for the player to decide on their own whether Hackett was operating with or without the direction/approval of the Alliance brass. (That Hackett is willing to act outside Alliance authority is evident if the player does the Renegade mission involving killing Darius and, again, selects certain lines that reveal this and, possibly, how they "apply" the debrief at the end of Arrival.)
I don't think Hackett would reasonably consider sending Liara around to find a solution to the Reaper threat prior to her becoming the Shadow Broker in ME2. Liara is portrayed as being a "very inexperienced" Prothean expert in ME1. She is young by the standards of her species and her work is being largely dismissed by the Asari. Her loyalty would probably also be in question with anyone outside Shepard's immediate crew (due to the activities of Benezia before she was killed).