No, long hauls generally don't carry more. Usually the minimum plus some contingency.
That said, the international requirement itself is a bit different. being flight to destination, + then add fuel for 10% of the time required to destination (different than just 10% of the fuel to destination due to weight change at arrival), plus fuel to the most distant alternate, plus 30 minutes holding fuel at 1500' above that alternate.
To reduce the 10% value most long flights are "rereleased" enroute so the the 10% factor is less, since the 10% can be calculated from the rerelease point. We are therefore initially released to a point short of the final destination, then at some point close to where we'd have to break off and fly to that point, we are "rereleased" to the final destination. All of this does not involve ATC unless we have insufficient fuel for the rerelease, and then we must continue to the original "released" destination.
In the end, we land with about what we would with the destination plus alternate plus 45 minutes, or pretty close to that. Some extra fuel may be added for weather or ATC contingency, but usually the 10% values cover all of that. The exception is very short flights that are international with close in alternates. Most of us (the pilots) have personal minimums which we look for which short cuts around all of this. I don't really care that much about all these rules if I have enough to get to my destination, then to my alternate and some comfortable reserve amount (I like 45 minutes), with extra for weather I think may be a factor, etc.
One more note is that domestic air carriers are not required to have an alternate if the weather is above certain criteria.