Part of it will depend on the pattern, not just the fraction killed but the variance. Even with the existing time period in some cases whole villages vanished.
With a higher death rate, more villages will vanish. I suspect that mortality rates were higher in cities than in the country side. (Read somewhere that until a century ago, cities were net population sinks, subsidized by immigration from the surrounding countryside.) So a higher death rate will mean that large towns and cities will become more disfunctional.
The overall effect would be the collapse of anything like government.
Travel at the time was already hazardous. Take out 2/3 of the road side inns and more travelers spend far more nights under the trees. With governmental collapse, highwaymen have more freedom to act. Travel more than ever becomes a matter for armed caravans, and is no longer a family affair.
With fewer towns, there is less reason for such caravans to travel. Only villages that are on the path between surviving larger towns will see the caravans.
The tradition of the medieval fairs would vanish.
Each village or town has to become far more self dependent. Prices of anything that cannot be made locally skyrocket. Wooden and stone tools replace metal.
Language languishes. (couldn't resist) No, not languishes, but starts to diverge. Much of Europe becomes like Prussia -- a raft of tiny kingdoms, essentially city states: You could rule a radius 1 armed horseman's day's travel.
The rise of nations is delayed by some {insert plausible time here}
Population that survived the plague might consolidate into larger villages, leaving even larger pockets of empty space between.
Because it would be semi random, there would be paths and streaks of people left. You can model this if you like with a large hexagonal grid. There is a known (but I can't cite references) about the distribution of cities, market towns, villages and hamlets based on travel effort costs. It's your typical 80/20 rule all the way up. E.g: 80% of the people live in hamlets. Of the remainging 20% 80% (16% of the total) live in villages, ... hamlets are about a day's round trip apart. In a one dimensional form you get something like
h-v-h-h-M-h-h-v-h-C-h-v-h-h-M-h-h-v-h.... With of course major perturbations: Any place that roman roads cross will become at least a market town. Ditto a bridge or ford over a river.
After the big die off, you will have connected lumps of varying size and shape. Some under some critical size will either die off and/or lose population to immigration to successful towns.
For story building, taking one such medium to large collection you can have fun with internal conflicts and struggles for dominance both internally and externally.
Don't forget the monasteries. In many cases these were located at the end of nowhere, and they may have tended to better survive the plague, due to isolation, and generally better health.
As a thread: Due to the depopulation, there may be exhortations to have children. The church (at least locally) may abandon the requirement of celibacy in the clergy.
Which brings up another issue: Bishoprics would have been isolated. Re-establishing contact with the papacy could prove interesting.
Commerce would redevelop along rivers first -- a much smaller group can be defensively independent on a river, and the cost per ton mile is lower.