I think the biggest mistake people make when thinking about the next step is to want it IMMEDIATELY. "Hey, I woke up today, and want more money. Where is it?"
This also is a reason that the fastest way to make more money is to switch jobs: the other place has the immediate need, and resources.
The down side of that is that you take a risk on the new organization's culture, and have to start building your network from scratch.
If you like the place you are, start the conversation now, but think in your mind about action next year. Why? Because budgets are already set. Your timeliness is a good one, to prepare for next year's budget. If you force the issue this year, you are making the choice twice as hard on your boss now: they have to take the money away from something else, in order to keep you. As good as you are, that's no favor.
If you communicate your expectations and give them the evidence, they can go to bat for you during the budgeting and HR planning cycle. If they don't go to bat for you, well, that speaks louder than any happy talk.
If this pace is too slow for you, then don't join a big org. And certainly, don't be in government. (Or a university!)
It does seem like you aren't stretching very far to explore your possibilities. These two are just the ones that have fallen in your lap. And far from ideal. What positions are open now, foe the money you want? Would you take them? That's worth putting in some time to research.
And, by the way: what is your next step after this one? It's not to early to aim towards that, if you have something clearly in mind.