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How many games into the season are you?
This sounds like it might just be an issue with sequencing, meaning that what really matters most in terms of run scoring is when good outcomes happen, not just that that good outcomes occur. I'm realizing I'm not explaining this well. But basically it matters most that hits and/or walks are clumped together in an inning and not just scattered throughout a game. Obviously, the more good hitters you have getting hits the better the chance they combine these in meaningful ways that lead to runs.
Upshot is, I'm thinking this is a small sample size issue that will correct itself given time.
Then again, you've mentioned hits and batting average but not whether your team has a good OBP or good slugging. If the hits are mostly singles, and if you don't have many players who walk often, and especially if those things are true but you don't have great team speed and maybe you have guys who hit into double plays, well, a lot of hits might not lead to a lot of runs.
What can you do about it? Probably just wait a bit for it to self-correct. Batting order can play some role, but most evidence seems to indicate it doesn't make a very big difference, though every little advantage counts. (For instance, I try to make sure my number 2 guy in the lineup isn't someone prone to hit into a lot double plays.)
(Now, re-reading, I am reminded that you say this is a White Sox rebuild, and I'm guessing that means using the current White Sox team. So, yes, the answer might indeed be the "not being very good" thing you said above. Then the trick is to identify why your offense isn't very good and try to fix it. Which hopefully means some of what I typed above is still relevant and slightly helpful.)
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