Getting a promotion - how do I go from underpaid in my previous role to overpaid in the new one?

I was hired as a Jr Sys Admin a year ago at less than the average starting salary. Since then I have been the hardest worker in the room and have pulled as much, if not more, weight as the (non-Jr) Sys Admins I work with.

I spoke with my boss about a potential raise and we talked about a promotion to a Network Engineer role that has been a position of need for them. I also made it clear that I was interested in this role as a long term career, so I'm not exactly doing them a favor in that sense.

I won't be a Jr anymore so it is a promotion, but in terms of the amount of work I've already taken on, it is a lateral move. I know I will be able to comfortably handle my new responsibilities.

I am anticipating being offered a ~10% raise, which would again be underpaid at the new role.

How can I argue for ~20%, which would be a huge jump but still only slightly above average for the new role?

Thanks.

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