You might like to look at 3 PAYG - very competitive PAYG prices (3ppm calls, 2p text, 1ppMB data). Some calls that are free on GG PAYg (e.g. 0800) are chargeable on 3. 3 also have bundles that you can buy on PAYG. They have an interesting (and seemingly good deal) oversea package for 18 countries - see their site for more details.
I'm currently with GG (and would definitely stay with them if it wasn't for the 3 pricing), have recently got a 3 PAYG SIM and will be trying out the overseas package when I'm in the US for 3 weeks soon.
My grand daughter (a Uni undergrad) has been using Gif Gaf on a rolling monthly contract at £10 a month for, I think, 500 phone minutes and plenty of data and texts. Her parents have followed suit. She bought a used unlocked iPhone 5 (32 gig) from Game with a 1 year guarantee.
They use O2 indeed, and all 3 are happy with Gif Gaf.
They looked at 3 but were concerned with the coverage. With an unlocked phone one could have different PAYG SIM for overseas us, I think, although the number would be different. We all have an extreme aversion to 1 to 2 year contracts as it is difficult to change suppliers.
The best deal for an international SIM that I've come across is Travel Talk SIM and it is one of the ones recommended by Money Saving Expert - roaming. I've had one for a few years and a lot more convenient than faffing around buying a local SIM in each country. It used to be a lot cheaper using it in Europe (as well as elsewhere), but as the EU has forced down prices it doesn't have quite the same edge as it used to.
I've used it in the US (and Europe and Jordan), but as their countrywide partner in the US is T-Mobile the coverage is particularly poor (T-Mobile seem to only cover urban areas in the US). By way of contrast, 3 use AT&T in the US (with far better rural coverage) - which is what prompted me to consider alternatives (I did consider buying a US throw-away phone, but decided against it - partly as I managed to pick up a Moto G 4G (1st generation) in Tesco for £89 about 4 weeks ago and could not resist.
I have recently changed from O2 PAYG to O2 PAYG with £10 Big Bundle. Below is a comparison O2-giffgaff. Being a lightish user I will see how I get along with usage, using the O2 £10 Big Bundle,then change to giffgaff if I start to use more than my allotted 100 minutes,1000texts 500MB of data. Once again thank you for your help/advice.
£10 Big Bundle
• 100 minutes,
• 1000 texts
• 500MB of data
£10 per month
• 500 UK minutes
• Unlimited UK texts
• 1 GB Internet
• Free to giffgaff numbers
If you're a light user then rather than buy a bundle that expires every month you'd be better with a straightforward PAYG that doesn't expire (usually requires making 1 call every 6 months though).
I currently have a 3 PAYG SIM with the 3p per minute for calls, 2p per text and 1p per MB deal as mentioned above. I believe this is the cheapest PAYG deal available. The call credit doesn't expire.
I also have a giffgaff PAYG SIM in the mobile but have temporarily disabled it as they seem to spam me with texts encouraging me to use it. PAYG deal on that is 20p per minute and 20p for 20 MB per day and again the call credit doesn't expire.