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A share secure loan through a credit union is usually what's suggested. May want to start here:
I didn't know that was even a thing. I've requested CLI's before without having a personal Loan. However, I did have a Auto loan.
Having an installment loan can round out your profile, but it shouldn't hinder CLI's should it?
If a person spends the time building up their CC base/relationship, they should see increases.
If you are considering a loan, I agree that a CU would probably be a better option than most other lenders. Since you have a Discover it, you might fill out their pre-qual. They also offer loans with no fees.
If our OP decides to explore adding a loan, he should go to the link given by KIB. The solution is not merely getting a loan. There's the whole question of what you do with the loan after you get it. That's the reason it is called the Share Secured Loan Technique. The only three lenders we know of that are compliant with the full technique are given in that thread. It would be great to find a few more.