In theory advance tickets are sold 12 weeks ahead (when the actual timetable should be finalised allowing for engineering works etc), however with the ongoing disruption to various operators and general Network Rail incompetence not all of them are achieving this at the moment so it can be a bit of pot luck.
There are varying numbers of advance tickets at different price bands for each train; when the cheapest is sold out the next most expensive is offered and so on. Busy trains will have fewer or none of the cheapest tickets. In theory additional cheap rate tickets will automatically be made available if a particular train is not selling well, but the level of demand usually makes this unlikely so it is unwise to expect it.
It is worth checking earlier/later trains if you can be flexible as some will often have cheaper seats than others. Avanti West Coast has a useful
Cheapest Fare Finder which shows the prices at different times/days if you are very flexible.
It can also be possible to get cheaper rates by splitting the journey into shorter sections, normally staying on the same train. So far as Preston is concerned however, it would usually mean changing at Crewe as through trains would normally be at the same rate. But for example if the Liverpool train still had cheaper seats available, you could book that to Crewe and then a separate Crewe - Preston ticket if that is cheaper than the through fare. You can work it out yourself on the relevant train company websites or there are a number of split fare websites which will do it automatically but charge commission on the saving. Trainline have also just started one but I boycott them as they charge too much for ticket sales! Allow plenty of time for connections though. And it is obviously better to bag the cheap tickets for your actual journey if you can.