I present just a couple of ideas that might make a bridge between the discrete and continuous.
Firstly, one thing I notice with many students is they don't have an idea that the probabilities of all the outcomes add up to 1. This stops them being able to do a lot of calculations for continuous distributions later. Moreover, it means they don't get the fundamental idea of distribution which is a description of all the possibilities and how likely they all are.
I think that we should probably never ask for the probability of any event in isolation but always also ask for probabilities for a set of events that complete it to the whole sample space. For example, don't just ask for the probability of getting a 7 on two dice, but also less than 7 and more than 7. At the very least, always ask for the probability of A and not A. My instinct is that to start with, students will calculate both based on the ratios, and maybe later will realise there's an easier way. But at least we'll drive home the idea that events never exist without other events to fill them out.
Secondly, perhaps a road to asking about continuous probabilities, is to ask about a discrete set of objects with continuous measurements on them. If the measurements are written with decimals it will be even more obvious that they are supposed to be continuous.
For example, take 14 trees with their heights measured in feet:
60.92 63.39 64.10 59.07 63.05 59.64 60.07 60.69 60.28 61.62 58.49 56.81 56.43 59.49 (from the Loblolly dataset in R).
Then you could ask questions like the probabilities of choosing a tree with height in the ranges 55 to 57.5, 57.5 to 60, 60 to 62.5 and 62.5 to 65 including drawing a graph showing these probabilities. Asking various different collections of zones might help to necessitate the axis being marked out on a scale rather than in discrete labels. Choosing zones that aren't equal might make it easier to necessitate a way to show the amount of probability in a way other than height so that the graph isn't so misleading.
A dataset with 100 numbers in it might make this even more obvious and lead towards the probability distribution which has infinitely many numbers.