Calculating confidence intervals for ordinal logistic regression predictions

I'm looking to plot the predicted probabilities for an ordinal logistic regression for a 3-level factor with confidence bands around the prediction lines. I'm struggling with the proper way to calculate the confidence intervals for all three predicted probabilities. Here's some of the output I'm working with:

Predicted Logits

  logit(P[Y<=1]) logit(P[Y<=2])
1     0.09022718      0.9875311
2    -4.13627553      1.4147613
3    -0.16576391      0.7315399
4    -4.39226662      1.1587702
5    -0.42175500      0.4755488
6    -4.64825771      0.9027791


Standard Errors of the Fit

  logit(P[Y<=1]) logit(P[Y<=2])
1       2.324888       2.332632
2       2.654493       1.389083
3       2.109122       2.110436
4       2.654716       1.329636
5       2.004890       1.999742
6       2.678278       1.298629


Calculating the interval around the first probability is straight forward using the first logit ($1.96*SE_fit$), which I then convert both the logit and the interval limits to probabilities for the first level.

The second two probabilities are where I'm getting hung up. Calculating the estimated predicted probabilities is straight forward. The third probability is the 1 minus the converted second logit, and the second probability is 1 minus the sum of the other two. However, I'm unable to figure out how to calculate the confidence intervals around these estimates. How would I go about this?

For reference, I'm using the VGAM package in R with the vglm() function. Many thanks for any insight given.

Edit: For clarity, I am looking for confidence intervals, not prediction intervals.