Ideally you should be getting a new cable. The one you have is guaranteed to work with 125 VAC. It probably will work at twice that voltage, but it is NOT guaranteed to do that because it was not designed with insulation suited to the higher voltage. For this purpose, you might find you have two choices. If you look around for a power cord that looks just like the one you have with the same connectors on both ends, BUT is labelled for use with 250 AC, you can use that with an adapter like you've shown. All the adapter really needs to do for you is convert the blade configuration of your cord's plug to match the socket holes in the country where you go. The other option would be to find a cable rated to 250 VAC with a connector on one end that matches the input socket on the back of your PSU, and a plug on the other end that matches the type used for the wall sockets where you will be traveling. Then you won't need an adapter.
BUT you also need to consider what the PSU itself says on its label. Many are designed to work with any voltage from 110 VAC up to 250 VAC, and with 60 Hz or 50 Hz frequency. Such a PSU can be used in a huge number of countries as long as you have the right power cord. But some simpler units are for one voltage only, so look at the label closely. In the past, many older designs were made with a selector switch on them, and you had to set it manually for the voltage you were going to plug into. But that is less common now.