What I do is edit the SimGameConstants.json to increase the "MaxContractsPerSystem" (usually to about 10) and then change the "ContractDifficultyVariance" to 8. Normally it's a 1 (meaning half a skull variance from the system default) Putting it to 8 or 9 really just means that any system can have any difficulty contract at any stage of the game. The increased number of contracts helps by ensuring that there will be at least a couple that should be doable, regardless of your lance composition. It also means that early on you can try a 5 skull mission and get thrashed, or get lucky and find yourself salvaging heavier mechs than what you'd normally get. I'm fine with that, as I don't really care for artificially hampering what I can get based on my campaign progression.
As a note, you can change this at any point in a career, it'll just take some in game time for the contracts to expire and renew to their newly adjusted levels.