Hi,
Not expecting Billy to come by with his friends, Jane has gone out to the grocery store over an hour ago.
'Has gone' is not good in this sentence because you also have the time expression 'over an hour ago'. The present perfect relates the past to now, but the time expression just relates the sentence to a point in the past. So, these two different things don't fit well together.
Yes, you could say 'had gone', because that relates the past to a point in the past. However, you'd need a context like this.
Billy visted Jane at 10am yesterday. Not expecting this, Jane had gone out to the grocery store over an hour earlier.
Note you'd also have to change 'ago'.
Write again if you have more questions, OK?
Clive