Matching Values and Copying to A new sheet.

Yep - here you go: just make the first paste of the first two columns conditional on booMatchFound being false and remove the sort at the end.

Option Explicit
Option Base 1

Sub MatchAndPaste()
Dim strPartsSheet As String, strDetailSheet As String, strSummarySheet As String
Dim intFirstDataRow As Integer, intLastDataRow1 As Integer, intLastDataRow2 As Integer
Dim sht As Worksheet
Dim strPart As String, strDetail As String, strOrder As String
Dim intCount As Integer
Dim booMatchFound As Boolean
Dim i As Integer, j As Integer

    'setup/intialise:
    strPartsSheet = "Sheet1"
    strDetailSheet = "Sheet2"
    strSummarySheet = "Sheet3"
    intFirstDataRow = 2
    
    intLastDataRow1 = Worksheets(strPartsSheet).Cells(intFirstDataRow, 1).CurrentRegion.Rows.Count
    intLastDataRow2 = Worksheets(strDetailSheet).Cells(intFirstDataRow, 1).CurrentRegion.Rows.Count
    
    For Each sht In Worksheets
        If sht.Name = strPartsSheet Or sht.Name = strDetailSheet Then
            Worksheets(sht.Name).Select
            Worksheets(sht.Name).Range("A1").Sort _
                key1:=Worksheets(sht.Name).Columns("A"), order1:=xlAscending, _
                key2:=Worksheets(sht.Name).Columns("B"), order2:=xlAscending, _
                header:=xlYes
        End If
    Next sht
    
    intCount = 0
    Worksheets(strSummarySheet).Cells.ClearContents
        
    For i = intFirstDataRow To intLastDataRow1
        strOrder = Worksheets(1).Cells(i, 1).Text
        strPart = Worksheets(1).Cells(i, 3).Text
        strDetail = "NO MATCH"
        For j = 1 To intLastDataRow2
            If Worksheets(1).Range("A" & i) = Worksheets(2).Range("A" & j) Then
                If Worksheets(1).Range("B" & i) = Worksheets(2).Range("B" & j) Then
                    strDetail = Worksheets(2).Cells(j, 3).Text
                    intCount = intCount + 1
                    If booMatchFound = False Then
                        Worksheets(strSummarySheet).Range("A" & intFirstDataRow + intCount - 1).Value = strOrder
                        Worksheets(strSummarySheet).Range("B" & intFirstDataRow + intCount - 1).Value = strPart
                    End If
                    Worksheets(strSummarySheet).Range("C" & intFirstDataRow + intCount - 1).Value = strDetail
                    booMatchFound = True
                End If
            End If
        Next j
        'if you get to here, you've cycled through the current record in sheet1 and attempted to
        'match it against all the records in sheet2 and found no match; in this case, need to
        'copy the Order and Part Number to the Summary sheet:
        If booMatchFound = False Then
            intCount = intCount + 1
            Worksheets(strSummarySheet).Range("A" & intFirstDataRow + intCount - 1).Value = strOrder
            Worksheets(strSummarySheet).Range("B" & intFirstDataRow + intCount - 1).Value = strPart
            Worksheets(strSummarySheet).Range("C" & intFirstDataRow + intCount - 1).Value = strDetail
        Else
            booMatchFound = False
        End If
    Next i
    
    With Worksheets(strSummarySheet)
        .Columns("A:C").NumberFormat = "@"
        .Columns("A:C").HorizontalAlignment = xlRight
        .Cells(1, 1).Value = "Order Number"
        .Cells(1, 2).Value = "Part Number"
        .Cells(1, 3).Value = "Detail"
    End With
    
End Sub