To be honest, they'll reach out to you, not the other way around - budgies are flock animals, and by gaining their trust, you're not only getting them to feel comfortable you but they are also going to want to spend time with you and include you in their flock.
When you spend as much time with them as possible and talk to them, interact with them, etc. outside of the cage, eventually you'll see that they're moving around normally in their cage, acknowledging you, and they may even come up to the cage bars to "talk" to you or see you more closely.
Personally, I remember when my girl was fully comfortable with me outside the cage - up until then, I had spent a ton of time with her. I would go down early in the mornings just so we could wake up together, I'd put on soft music and talk to her as much as possible, I'd read to her, I played the violin for her, etc. and I also would put my face closer to the cage bars over time. One day, I was looking at her and talking to her through the bars, and she hopped onto the cage bars from her perch, she was just clinging there, looking at me calmly. We were so close her little foot brushed against my nose. I just remember being so proud of her, and so excited that she was ready to continue getting to know me.
It may not happen exactly like this for you but you'll definitely be able to tell when they're comfortable with you outside the cage. A good sign is when they're moving around a lot, playing with their toys, chirping at you, and being interested and curious about you instead of wary.
I hope that helps
Best of luck!