American Airlines is relaxing part of its pet policy to let owners bring their companion and a full-size carry-on bag into the cabin.

Until this week, people who carried a pet into the cabin — which involves paying a $150 (£119) fee — could only have one other small item that fit under the seat.

Anything bigger, such as a carry-on bag with wheels, needed to be checked — for a $35 fee, or they could put the pet in the cargo hold.

Now American is letting passengers bring a pet in the cabin and also bring a regular carry-on bag or a personal item — just not both bags.

The old policy struck some pet owners as unfair, since they were already paying a pet fee.

The airline’s website now states that passengers “can bring one pet carrier as a carry-on if you pay the carry-on pet fee, [and] your pet stays in the pet carrier and under the seat in front of you the entire flight”.

Passengers can only bring one more item on board with their pet – either “a personal item like a purse or small handbag; or a carry-on bag that meets requirements and fits in the overhead bin”.

The destinations that accept pets include the 48 contiguous United States, Alaska, Puerto Rico, Canada, Mexico and some of the US Virgin Islands.

An American Airlines spokesperson told The Independent that the changes came into effect on 28 March, and they were made “to provide a more convenient and comfortable experience to customers whose pets fly American”.

Other US airlines operate similar policies to the one that American have just discarded. The Delta website states that “pets in kennels will count as your 1 carry-on item. In addition to the kennel, you are permitted to bring 1 personal item on board the aircraft”. The airline’s pet fee is $95 (£75) for domestic flights and $200 (£159) for international flights.

On United, pets can fly for a $125 (£100) fee, and count as one piece of carry-on luggage.

Conversely, both Alaska Airlines and Southwest already allow passengers to travel with one pet carrier “and a regular size carryon bag” or personal item. At present, some US airlines are refusing any animals to travel in the hold of their planes, except those that belong to re-assigned military personnel.

Rules in the UK and Europe are slightly different to the US. On UK airlines, including easyJet, Ryanair, Jet2 and British Airways, no pets – other than recognised service animals – are allowed in the cabin. Pets can be transported as cargo on BA flights, while the two low-cost carriers only accept service animals, and only on specific routes.

The policies on major European carriers vary. On Iberia flights pets weighing under 8kg can be brought into the cabin, though there are different restrictions for certain destinations – for example, animals cannot travel in the cabin or hold on any flights to London, while Brazil “only allows the entry of birds that meet a series of requirements”.

On KLM and Air France, pets under 8kg can travel in the cabin to certain destinations, but their carriers count as a piece of luggage and therefore incur “an additional fee” dependent upon the route and ticket class. The same policy applies to Lufthansa flights.