The phrase "to continue the run" means to keep doing something successfully or without stopping. It suggests maintaining a streak or series of accomplishments. Full definition
If Spurs are to win or grab a point they'll have to rapidly find their feet in front of goal and continue their run of defensive form in the league. (justarsenal.com)
Chrome normally continues running in the background, even after you close it. (howtogeek.com)
It's most certainly started as a productive year and I plan on continuing this run as long as I possibly can. (eleventhandsixteenth.com)