Do I just need a adapter?

You should be ok but I found my dual hair appliances got very hot. I had to use my hair dryer on low settings.

Make sure the plug adapter is designed for higher amperage/high watt appliances. It should note somewhere it will work with hair dryers and similar items.

I usually suggest the adapters sold here on the RS store, but they are designed for low power items like phone chargers so you will need a higher capacity one.

My wife has carried a Conair gas powered curling iron--with small gas cartridges--last 20 years. Works great.

We picked up a 220 volt USB transformer 2 weeks ago in Madrid for 6 euros for tablets and phones. Not using the old style 220/110 volt adapter any longer.

Perhaps you could wait at buy one at your first destination in Greece.

There’s a mismatch between the adapter and your iron.

The adapter is stamped 6 A 125V - 250V

The iron goes from 100V to 240V. Amperage unstated.

That said, you’re not going to see 100 V in Europe so you won’t be using the adapter except under 220 V.

As others have said, the heat output is different under different voltages.

What sort of plug does it have? Annoyingly none of the photos show the plug, and it doesn't say.
Is it 3-pin or 2-pin. If it is 2-pin, that adapter is OK, if it is 3-pin you need a 3-pin adapter.

Not being a person who has ever curled their hair I cannot say whether these items normally have 3-pin plugs, as they have mains electricity in proximity to the human body I would hope they are earthed (3-pin).

First, the two pin (Europlug) adapter is limited by European electrical codes to 2½ amps at 230V, or 575W. Check the wattage on your iron. If it does not say less than 575W (or doesn't say anything) you should not use it with a 2 pin adapter.

Second, it probably has a "polarizing" US plug (one blade wider than the other). UL requires the polarizing plug to ensure important safety conditions, but almost all European power is not polarized. In other words, if your appliance has a polarizing plug, it's not UL approved for use in Europe. (Swiss power is polarized, but only if used with a grounding adapter, which the one you show is not.) So you cannot legally use that iron in continental Europe.

Saying that, I know you will ignore their codes and use it anyway, but at least make sure it is not left plugged in. That is a fire hazard.

My wife has carried a Conair gas powered curling iron--with small gas cartridges--last 20 years.

Sounds like a good solution, as long as you can get the gas cartridge over there. I doubt that you can take one on a plane.

Can say anything about your variants I'm using Babyliss one which I found here
It's ok for US and EU

The adapter is stamped 6 A 125V - 250V

I am guessing these products are made in some 3rd world country, and there is no regulation on what they are allowed to claim on them. But in Europe, two pin adapters such as that one are limited by code to 2½ amps, so it would be technically "illegal" to use it for 6 amps, and it should not be recommended for that on this site (per the guidelines).

There’s a mismatch between the adapter and your iron.

No mismatch. Although the manufacturer probably doesn't even know what that means, if he did, it would only mean it is rated up to 250V (the 125V is irrelevant), and it can conduct up to 6 amps. The input for the straightener is within the voltage limits.

BTW, the page for the straightener has a misstatement, it is not dual voltage. Dual voltage means it operates on just two voltages with a ratio of 2:1 (230V and 115V, for example). A switch is used to put the resistive elements, which are designed for the lower voltage, in series for the higher voltage.

This device is multi-voltage. It takes the input voltage, whatever it is, more than 100V, and limits the output electronically to a lower voltage, probably in this case 100V, which is what the resistive elements are designed to work at. This is the way all USB voltage converters (chargers) work.