Is there any loan history from the past that is not showing up on the credit reports that is positive? If not, I would ask that the history be reported. That won't cost much.
Do some research on the shopping cart trick. There are quite a few cards that can be obtained without a hard pull. However, I would be reluctant to add more than 2 cards every 6 months. The problem with adding new accounts is that it dilutes the average age of accounts. However, if there is a strong average age of accounts, opening a new card will not be so damaging.
Credit scores can be rehabilitated relatively quickly (6 months) if a consumer is willing to put in the effort. If it were me, I would order myFICO reports for all 3 bureaus from myFICO. This is the only place to get the full FICO analysis of what helps and hurts one's score. I would not use CreditKarma for advice on credit repair, because the algorithms that it uses are fake and don't reflect real FICO scores. These Credit Karma numbers are wildly off in situations where there are no active credit accounts (such as what you're suggesting).
In my experience, the fastest way to get a huge bounce from FICO is to open one credit card account and find out when that credit account reports to the actual bureaus (this can be done with a daily puller). Then, I would make sure that the utilization of all open credit card accounts is between 1-3% of the total balance. I would make sure that every month I paid so that exactly 2% of the balance shows on the credit report. Do not pay off the account to a zero balance every month--this will hurt FICO scores. Just pay down to 2%. What you will find is that at the 6 month point after the account opens, you will see a large bump in FICO scores. Assuming that the utilization reports around 2%, a consumer can expect to see a large bump in scores (from 50-100 points).
Once a consumer vaults into the 650 range, there are many more credit options that become available. From there, it can take another 6 months to push into the 700+ range where rewards cards may be worth exploring.
Meanwhile--while that new credit card is marinating 6 months at a time--I would be looking to surgically remove select negative TL's through the FCRA and arbitration with the CRA's. The place to start is to pull hard copy reports (not the online reports) by ordering each report from each bureau so that they mail an actual hard copy report to your home.