Dankelmann,　Guo　and　Surmacs　proved　that　every　bridgeless　graph　G　of　order　n　with　given　maximum　degree　Δ(G)　has　an　orientation　of　diameter　at　most　n−Δ(G)+3　[J.　Graph　Theory,　88(1)(2018),　5-17].　They　also　constructed　a　family　of　bridgeless　graphs　whose　oriented　diameter　reaches　this　upper　bound.　In　this　paper,　we　show　that　G　has　an　orientation　of　diameter　at　most　[Formula　presented],　where　g(G)　is　the　girth　of　G.　Moreover,　we　construct　several　families　of　bridgeless　graphs　whose　oriented　diameter　attains　[Formula　presented],　and　prove　that　the　upper　bound　is　tight　for　Δ(G)≥4.　We　also　give　a　necessary　condition　for　a　bridgeless　graph　to　attain　this　upper　bound.　Furthermore,　we　verify　that　if　G　is　a　3-connected　graph　with　girth　at　least　5,　then　the　oriented　diameter　of　such　G　is　at　most　[Formula　presented].　©　2022　Elsevier　B.V.