Well considering that Lilly is said to have two "different" dads I have this weirdest feeling that she does have two dads. Now don't get me wrong i'm not insinuating that Larry is gay, but that Lilly has fabricated him as her father figure, and he has decided to go along with this facade Since her original father was so dear to her, Larry decided to take over for her safety, so when she lost Larry she lost her last lingering hope of her father's memory.
Now assuming that the outbreak had been at least a few days in this could be a big possibility as her father could've died within the first few moments of the outbreak sacrificing himself so she could get away, while Larry's family could've died on the first few moments by being bitten when the virus was relatively new.
Now a lot is up for debate as it's up to interpretation if this could be true a few things I found a bit weird: Larry's treatment of Lilly, The mentioning of Lilly's mother and the ring, Lilly's lack of knowledge about Lee's criminal past(I'd assume Lilly and Larry were living close or with each other), lack of mentioning of any history of Larry's previous history of being a military man or being married, and Lilly's lack of knowledge in terms of treatment for her dad's illness and some qualities of Larry other than he's hurting.
Now like I said this is all speculation and only based on opinions and the recent fact that in canon Lilly's dad is named Everett Ray Caul which he could go by Larry but that seems like an awful stretch. And i'm not really 100% on this theory but it does seem like an interesting turn if it is chosen.
TLDR; Larry is Lilly's fake dad her real dad is Everett who died after sacrificing himself for her.