"Best way..." depends on your definition of "best."
However you decide to implement the change, the first (and probably most important) thing to do is to decide what you want, and then go find the details of an alternative flight that meets your criteria, note the flight number, times, route, details, etc., then call United and spoonfeed the agent with the details. You should not be charged anything (this assumes you have not accepted their proposed alternative). This is straightforward and should result in what you want. The only downsides are 1) you have to do the work finding an alternative routing/flight that you want (only you can/should do that, you've already seen what their computer thinks is best for you), and 2) you probably have to wait on hold (maybe for a long time).
You might be able to make the switch yourself online, but I would not trust that. I'd want a human to make the change for me, read back the details to be sure they got them right, then send me an email confirmation.
The primary rules you have to follow are these: Since it's a change form what you originally booked, you have the opportunity to decline their proposed alternative, and choose something that you prefer. But that alternative needs to be (all of the following):
- Routing must begin and end in the same cities as your original booking. It can take a different route on the way (different connections, fewer/more connections) but start and end points can not change.
- It must depart on the same day as your original booking (they may bend this rule if there are no reasonable options).
- It must be in the same class of travel (eg: economy) as your original booking (don't expect an upgrade)
- The cost of your new, requested itinerary does NOT need to be the same or lower than what you paid...if they're selling tickets for more $$$ now (say, on a nonstop) they should not charge you anything extra, you already paid them to get you from Point A to Point B; they now say they can't get you there on your original flight, so charging you more for a different (better) way of getting you there is not fair. If they say it'll cost more for your preferred alternative, ask to speak with a supervisor.
So, fire up the computer, start searching for what's available now, find what you like, call them and ask for that. Be polite and friendly, but don't be shy about asking for what you want.
Of course, things may change again later, but all you can do is act based on what currently appears to be available.
