First and foremost: Where are you getting your scores from? If they are from Credit Karma, they are not your real FICO scores and you should ignore them. You can go to CreditCheckTotal dot com and get all 3 of your FICO scores for just $1. It's a 7 day trial, so be sure to cancel before the 7 days.
Secondly, I don't understand why your wife wouldn't want you to get a secured card. There is absolutely nothing wrong with getting one! They are there to help people build/rebuild their credit. I would check out either Discover, Capital One, or maybe a local credit union. Discover and Capital One both have the pre-qualify links, so test those out before you apply. The best part is, after 7ish months of responsible use, they will graduate your card to a real credit card, and return your security deposit. You don't need to give them a lot, it's usually only $200, and that would be your limit.
There is always a possibility of increasing your scores. I was able to get mine up from 580 to 700 within a year (not even a full year yet). Do you have any other derogs on your report like late payments or collections? If so, those could definitely be hurting your score. If your scores are so low just because you don't have much credit, that's an easy fix! Starting with a secured card is usually the easiest solution. I'll let some others chime in and give you more info, but that's my 2 cents for the day.
Inquiries falling are not going to give you that much of a jump. You are missing age and credit mix from what I gather. What type of account is the one for 440? If its an installment then you have that taken care of as long as you don't pay it off. Paying it off will bring your scores down right now. You need a revolver as in a secured or a couple to fix the mix.
I would suggest first that you get your real fico scores. You can get a trial at CreditCheckTotal.com for a buck and it will show you where you are at now. Then post your scores and your accounts with the balances and original loan amount, then someone can help you figure out your next step.
Even though you are a newbie... do you mind sharing the steps you took to get to where your scores are now? My scores are similar to what you started out with and it is amazing how your score jumped up like that. I just got the Capital One Platinum $300 SL; First Access $300 SL and Citi Secured $300 Deposit and was advised I should stop there with the CC and let time do its thing.