score:2
The exact answer here will depend on the exact "fare rules" of your ticket which obviously we don't have, but I'll go with the normal rules you'd see for most tickets on that route.
In general, when attempting to change a ticket like this AFTER the departure of the first leg, the prices that are used for the changed legs are those that were in place at the time when you first booked the ticket, which will generally be cheaper than the current fares.
When you do something like look on Google Flights, you're seeing "todays" prices, not those that existed when you first booked the ticket. It's possible these will be the same, but it's likely they won't be. By looking for "one-way" flights you're also seeing the prices for one-way flights, which are generally higher than the price for return flights that your ticket is most likely based on.
The only places that will be able to give you accurate pricing for a change is either United or (possibly) the travel agency that you purchase the ticket through - if it was purchased through a travel agency (and even then, the travel agency will likely need to work with United to work out the correct prices).
In theory, you could try using the "change flights" functionality on the United website, however the unfortunate fact is that this functionality is known to not always function correctly - it may not give you all options for flights, and may overprice what it does offer. As a result, this is generally not the recommended way to do it, especially for a ticket that includes at least 3 countries in it!
Which leaves you with only one option - call United. As you've said, their agents have access to systems that should be able to calculate the new prices relatively easily, although even then they can sometimes have difficulty pulling up the historic prices depending on the exact situation.
Upvote:1
You can presort with a tool like ITA Matrix https://matrix.itasoftware.com/search
Search for "one way", "calendar of lowest fares" and United as the airline (routing code "UA+").
If there are any dates that look better than others, you can start with these. For May this does not look promising:
Upvote:3
I would try the following:
For instance in today’s routing via IAH and ORD there’s availability down to K class, while tomorrow’s only goes down to W (which is 4 buckets higher than K). On Friday one routing is fully booked while the other only has availability down to V (one notch above W).
Once you’ve found something interesting, check on the UA site.
Fare rules are complex enough that this is not guaranteed to get you the best price, but it should be a step in the right direction. Hopefully!
Note for instance that your original fare could include restrictions such as a minimum or maximum stay, having to stay over a week-end, travelling only on certain days of the week, black-out periods, and more.