This script has some repetition, making it a perfect candidate for extracting functionality... Something like 'ClearAllFilters'!

add_to_destination_worksheet.vb

'This subroutine creates adds the filtered data from Sheet1
'to a previously-existing destination Worksheet (called "Destination" here)
Public Sub AddToDestinationWorksheet(StartDate As String, EndDate As String)

    Dim wksData As Worksheet, wksTarget As Worksheet
    Dim lngLastRow As Long, lngLastCol As Long, lngDateCol As Long, _
        lngDestinationLastRow As Long, lngDestinationFirstCol As Long
    Dim rngFull As Range, rngResult As Range, rngTarget As Range
    Dim varFiltered As Variant
    
    'Set references up-front
    Set wksTarget = ThisWorkbook.Worksheets("Destination")
    Set wksData = ThisWorkbook.Worksheets("Sheet1")
    lngDateCol = 8 '<~ we know dates are in column H
    
    'Identify the full data range on Sheet1 (our data sheet) by finding
    'the last row and last column
    lngLastRow = LastOccupiedRowNum(wksData) '<~ straight from VBA Toolbelt!
    lngLastCol = LastOccupiedColNum(wksData) '<~ straight from VBA Toolbelt!
    With wksData
        Set rngFull = .Range(.Cells(1, 1), .Cells(lngLastRow, lngLastCol))
    End With
    
    'Apply a filter to the full range we just assigned to get rows
    'that are in-between the start and end dates
    With rngFull
        .AutoFilter Field:=lngDateCol, _
                    Criteria1:=">=" & StartDate, _
                    Criteria2:="<=" & EndDate
        
        'If the resulting range contains only 1 row, that means we filtered
        'everything out! Check for this situation, catch it and exit
        If wksData.AutoFilter.Range.Columns(1).SpecialCells(xlCellTypeVisible).Count = 1 Then
        
            MsgBox "Oops! Those dates filter out all data!"
            
            'Clear the autofilter safely and exit sub
            wksData.AutoFilterMode = False
            If wksData.FilterMode = True Then
                wksData.ShowAllData
            End If
            Exit Sub
            
        Else '<~ otherwise we're all good!
        
            'Store ONLY the visible cells (and skipping the header row), which
            'match the specified date range
            Set rngResult = .Offset(1, 0) _
                            .Resize(.Rows.Count - 1) _
                            .SpecialCells(xlCellTypeVisible)
            
            'Identify the last row on the destination sheet
            lngDestinationLastRow = LastOccupiedRowNum(wksTarget) '<~ <3 u VBA Toolbelt
            
            'Identify the first column on the destination sheet:
            '
            'If the last row in column A is empty, execute an xlRight from
            'there to find the first-occupied column
            If wksTarget.Range("A" & lngDestinationLastRow).Value = vbNullString Then
                lngDestinationFirstCol = wksTarget _
                                         .Range("A" & lngDestinationLastRow) _
                                         .End(xlToRight) _
                                         .Column
            Else '<~ otherwise, the data starts in column A (i.e. 1)
                lngDestinationFirstCol = 1
            End If
            
            'Now that we know our last row and first column, setting the target is a snap!
            Set rngTarget = wksTarget.Cells(lngDestinationLastRow + 1, lngDestinationFirstCol)
            
            'Append (i.e. copy to the bottom of the data range) the filtered
            'results to the Destination sheet
            rngResult.Copy Destination:=rngTarget
        End If
    End With
    
    'Clear the autofilter safely
    wksData.AutoFilterMode = False
    If wksData.FilterMode = True Then
        wksData.ShowAllData
    End If
    
    'Holler at the user, our macro is done!
    MsgBox "Data transferred!"

End Sub