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From the information you provided we would recommend looking into Mr. Smith's military pension application file. A pension file can be a good source of genealogical information though content can very from pension to pension. After a review of some online resources we were able to find a pension index card for an Alfred and Kitty Smith. From the form Mr. Smith applied for but never received a pension. His wife Kitty had better success and was approved for a pension some time after1892.
The majority of Civil War pension files in the Records of the Department of Veterans Affairs (Record Group 15) are in the custody of the National Archives at Washington, DC - Textual Reference (RDT1). Under normal circumstances you could request a search and order copies of military pension application files based on Federal (not State or Confederate) service during the Civil War. You would complete a NATF Form 85 and mail it to the address on the form. Researchers also had the option to come in to view the records.
However, due to the COVID-19 pandemic and pursuant to guidance received from the Office of Management and Budget (OMB), NARA has adjusted its normal operations to balance the need of completing its mission-critical work while also adhering to social distancing guidelines for the safety of our staff. As a result, NARA has closed facilities access around the country to both staff and the public, it has also suspended reproduction activities. (Please note that recently selective NARA facilities have allowed for reduced staff access. However, there may still be some restrictions to building access that may prevent some staff from providing a thorough response to requests that require us to consult the records at this time)
Another resource that may be of interest to you can be found within a National Archives pamphlet relating to the research of African American Genealogy: https://www.archives.gov/files/publications/ref-info-papers/rip108.pdf
We hope this information is helpful, and best of luck with your research!
The name of the the slave owner on the Freedman's Savings & Trust card, is Mrs. Didlake. In 1860 there is a "Mrs. Didlake" in Lexington KY that owned 1 mulatto male slave age 21 or 31 (hard to read the first digit). Robert J. Didlake owned 30 slaves in the 1850 slave schedule, some of whom were close in age to Alford. Robert J. Didlake was born in Clark Co KY and died 20 Feb 1859 in Lexington KY. His wife's name was Elizabeth. Although I could find entries in Fayette Co KY probate records on familysearch.org that his estate was being probated, none of the records that I found were either the will, which might have listed some of the slaves by name or the inventory of his estate that would have named all the slaves. You may want to contact the Fayette Co. KY probate court to see if the inventory still exists.
I found the will for Robert Didlake, father of Robert J. Didlake in Clark Co, KY Kentucky Probate Records, 1727-1990; https://familysearch.org/ark:/61903/3:1:33SQ-GP3V-WQG?cc=1875188&wc=37R5-JW5%3A173387701%2C173866401 The will was written in 1836 with codicils added in 1840. Best I can tell his estate was probated in 1842. There was no slave named Alford/Alfred in his will or inventory.
Alfred & Kitty (aka Catherine) Smith had a daughter named Mary Smith, born about 1856) according to the 1870 & 1880 Census. If she married someone with the surname Price, this would be her death certificate. Note that the names of her parents on the certificate are Alfred Smith & Catherine Daniels. If she did not marry someone names Price, it would not be her.