Logan had a perfect debut on Saturday, making a string of saves and stopping two penalties in the shoot-out as Hibs beat Dundee United in the William Hill Scottish Cup semi-finals.
And Oxley knows he will have to perform well when he returns from suspension against the Ladbrokes Championship title-winners at Easter Road on Wednesday.
Stubbs told Hibs TV: “Conrad’s come on a short-term deal and we obviously know he is working his way back to full fitness. He came in and did really well, no one can argue with that, but Mark is my number one.
“But what Conrad has done is certainly provide real competition for places with Otso (Virtanen) for that number one. Competition for places is huge in any team.
“Conrad is chomping at Mark’s heels, which is great that should spur Mark on, knowing that if his performances drop, Conrad and Otso are waiting in the wings to get an opportunity.”
Hibs and Rangers meet for the fifth time this season on Wednesday and will cross swords again at Hampden after Mark Warburton’s side overcame Celtic in the other semi-final.
The result means on-loan Celtic players Liam Henderson and Anthony Stokes can play in the final but Stubbs claimed he had no preference for his team’s opponents.
The former Celtic defender said: “I went to the semi-final on Sunday with the thought of going there to enjoy watching a game of football and, whoever we got, enjoy playing them on May 21. I didn’t have a preference.
“When I look at it now, the fact we are playing Rangers has helped because we can call upon Stokesy and Hendo, so from that point of view I am pleased that has happened, but I didn’t have a preference.”
Meanwhile, Stubbs revealed Farid El Alagui is expected to be out for about three weeks after suffering a hamstring injury at Hampden.
Liam Fontaine (thigh), Dylan McGeouch and Dan Carmichael (both groin) remain on the sidelines but James Keatings is back after a cup ban.