Most schools have a few airlines with whom they have contacts, perhaps a local airline or one whose chief pilot knows the school's head of training. Naturally the grapevine then helps. This business is all about who you know. This is done on an informal basis, and no school is especially endorsed by an operator for basic training (jet orientation courses perhaps). Schools have tried for years, and claimed special favour, but when I looked into the courses I could not find any claims that stood up to scrutiny; the airlines just don't want to make any commitments.
Any informal links change over time and are subject to the market. For Example now BA say they will only take integrated graduates from certain schools. When they were actually recruiting newly-qualified pilots they used to look at modular students from other schools (SFT for example, when I did my course with them).
The only "first-job" airline I know of which has a policy of taking only integrated graduates is Flybe, and I know for a fact that they have just taken two modular graduates! This shows what you will find out about recruitment in this industry - policy and rules are so flexible as to be almost meaningless.
There's no need to even work as a flight instructor. You only need 150 hours to start a modular CPL course now, and could have the licences to sit in the right seat with less than 250 hours, on a modular course! Costs in the UK, minimum of around £40-42k.
Best of luck!