Assess computer back up data for Outlook Exchange and recover prior version of .pst or .ost

I have a full backup of my laptop on an external drive including the entire OS and data as well as data itself. I was transferring to another pc and a set of emails for a time period got deleted from the outlook exchange files so when I opened the .pst version I was missing the emails for that timeframe. I looked on the backup and there are prior .ost and .pst files prior to the deletion but Outlook does not recognize them as Outlook data files as they have additional numbers and letters after the file extension. I need first an assessment of whether the data can be recovered and if these other files need opened, I need them converted so they are importable as csv etc from exchange for use as non exchange. I am interested only in the emails and contacts. I am not sure of how complex this is but believe it is mainly a quick assess and then file conversion issue and that is how I priced it. If it requires more than I will consider that in the price discussions but my assumption seems to be reasonable.