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There's a lot of commentary out there that Chase doesn't like thin files, Chase doesn't do soft pull autoCLI's, etc. And the datapoints for that do exist, however I think it's extremely profile dependent. I'd had a Cap1 secured card for eight months and no other history when I applied for a Chase Freedom, and it was instantly approved for $1500. Of course, the credit addict in me would have liked higher, but hey, at least it wasn't a $500 toy limit. I've gotten two autoCLI's since then, up to $3500 after around five months, and then to $4500 three months after that.
I think the choice here is whether to wait to get to a point where her profile supports a $5000 SL (CSP), or whether to just get a Freedom or FU now/soon and use it a lot until she sees a preapproval for the CSP. I would personally go for the latter and try to establish a relationship with Chase sooner rather than later, that way when she is ready to apply for a CSP she's a lot more likely to get more than the $5000 minimum starting limit.
And, she should be able to get a free FICO score from Discover.
Data point: I got my secured Disco in July 2016. Nothing else on my file except HP's. Got a targeted offer for the Chase FU in February and approved for $500 (after 2 days of patiently* waiting).
My Disco has since unsecured and I seem to already be liked by Chase. I'm hoping I'll be able to report an auto CLI after statement #6 when they get jealous of Disco. I'm planning to garden until then.
To the OP, I don't think I'd go for the trio yet. Go for the freedom (since rumor has it that it's going to be discontinued for new apps) and let that get through 6 statements. Check for offers (in-branch if possible) and go from there.