Often if I do not know the league but crave action I will simply fade(go against) a big line move. If it is a stable league (no fixed matches) and I see a huge movement to one side it can be profitable to rely that the bookmaker's original odds were more correct. Often a sharp handicapper will make a move to one side. People who follow the markets will all also bet that side because of the initial line movement. The line will then overreact to the herd of people betting that side. There is money to be made betting against the herd.
I do not know if I understood correctly how you bet. Does this mean that in your opinion, when we do not know the league, but it is stable league, it is best to bet on higher odds?