Hollis looked set to continue as understudy when Stuart McCall signed Faroe Islands international Gunnar Nielsen as a replacement but he was handed the gloves for the Europa League home game against Kuban Krasnodar and has now played six games despite his rival for the jersey saving a penalty in Russia on his debut.
After picking up the Cash Converters Player of the Month for August, Hollis said: "It's always nice to get personal awards but it's more important to get results for the team and keep on pushing us up the league.
"I've only played a few games. I have done nothing yet. I take one game at a time and see what happens."
When asked if he felt added pressure because of the success of Randolph and John Ruddy before him, Hollis said: "You don't really think that way, you just go and play the games and try to play the best you can."
Hollis, who will start against St Mirren on Saturday, admitted he found it hard to be on the sidelines for so long but he was determined to take his chance when Randolph left.
"It's been frustrating but there's no complaints because he was top notch," he said. "I just had to keep plugging away.
"There was nothing I could do because he was outstanding.
"I knew there was a wee chance if I worked hard during the summer and did okay, that's all I was thinking.
"It was just before the first European game when I knew I was starting. I didn't know through pre-season.
"It was really pleasing to start. It's better than playing in reserve games, playing in big games."
McCall insisted Hollis had earned his chance on merit rather than through just loyalty.
"If Gunnar had been well above Lee in training and in the games, then Gunnar would have started," McCall said.
"But I think I have got two very good goalkeepers. There's not a lot in them, they both perform well in training and in the games they have played.
"It was only fair to give Lee his opportunity and at the moment he has taken it.
"And he is being pushed all the way by Gunnar so it's a good situation to be in."
Motherwell face a St Mirren team without a win this season and with manager Danny Lennon under pressure from some of their fans.
But McCall knows he will get a tough game against a side who are unbeaten in five games against Motherwell.
"I think we are all under pressure," McCall said. "We have all got challenges. Danny took St Mirren to the best finish they have had in a long time and won a trophy. He has got a good squad there.
"Football can change so quickly. Look at the Scotland game - if we get beat in Macedonia then everything is totally different - it's doom and gloom and no progression.
"But the players put in a really good performance. St Mirren have the players capable of it and one good performance and they will be up and running. Obviously we hope it's not against ourselves.
"We have always had really good football games against them, tight games."