Okay @Fairfield - I'm adding possible ESP connection to my list of zany problem solving techniques!
@ethos92 - we tend to use a sliding scale of most likely to least likely in cases like this. Based on typical patterns of behavior from the time your ancestor's most likely scenario looks to me like: husband goes to war, falls ill, goes in and out of hospital without ever making a full recovery, dies in service at a time when he is not in a situation to have his death recorded (for instance, he is released from hospital and told to return to his unit but he never makes it and his death happening while he is between the hospital and his company leaves him without a record). Then his widow dies in the decade after the War, leaving her daughter to be taken in and raised by a close family member. As she has no husband at the time no one has money for more than a burial and she remains without a tombstone, so there is no lasting record of her death. Any and all of this would make sense and be similar to other stories from the time.
Scenarios like - he went west, he came home but didn't leave a record, etc. are possible but much less likely. Just as his widow remarrying but not keeping her daughter with her. That did happen, but it was less likely.
The fact that there were pension applications filed, even though not granted, does mean that the applications still exist. Those have a lot of information on them. Have you gotten a copy from the National Archives?
There are two resources that I would say to look at, if you haven't already.
1. Newspapers from that era published a lot of small town local news. As @Fairfield alluded to above, you may find a record of the wife's death by examining records of her family members. A simple line in the local news such as "Mrs John Smith was in Jonesville for the funeral of her sister last week" could be a valuable clue to the date of the widow's death. The daughter may show up in the paper winning an award in the spelling bee at school and you would know when she had transitioned to the new household.
2. Diaries and letters from soldiers serving with your ancestor likely exist. They might just have a clue in them. Both of these options take a lot of searching. But sometimes they do yield valuable results where other methods failed. There are collections online, some of which are searchable by place. Local history organizations often know where diaries and letters from their area are held.
I realize I am giving more work to do instead of an answer! But unfortunately that's often the best we can do.
I have checked newspapers pretty extensively, but haven't come across any other soldier's letters as of yet. That would be quite a find if I can! Thank you for looking at this and the helpful suggestions!