There are three separate issues here, viz the Business Names ACt 1985, "passing off" and registered treademarks:
You need to show the FULL name of a company on "all business letters, written orders for goods or services to be supplied to the business, invoices and receipts issued in the course of the business and written demands for payment of debts arising in the course of the business" [Section 4(1)(a), Business Names Act 1985]
http://www.legislation.gov.uk/ukpga/1985/7/enacted
As far as the Business Names Act (alone) is concerned, that would be satisfied if you put 'CarCo' on your letterheads but then appended ''CarCo Hire Ltd, trading as 'CarCo" to the bottom of the page.
However, entirely separate to the provisions of the Business Names Act, you'd need to consider whether the similarly-named company could take proceedings against you on the grounds of "passing off". (In the first instance that would normally be a 'cease and desist' letter from their solicitor but, if you failed to comply, they could seek a court order). They could only do that if a reasonable person might assume that the two businesses were one and the same, thus leading to a loss of trade (or a loss of reputation) for the other company.
Having similar, or identical names, isn't in itself a problem where there's very little likelihood of people getting the businesses mixed up. (e.g. there are loads of firms called A2B taxis across the country but it's unlikely that someone calling a cab from A2B in Exeter would assume that it's the same A2B that operates in Edinburgh. Similarly, there are probably lots of hairdressing businesses across the country using names like 'A Cut Above'. As long as there's no 'passing off', there's not a problem.
Lastly, you'd need to check whether or not the other firm had registered a trademark using their name. The reason that Toys R Us could keep preventing firms using names like 'FishnChips R Us' wasn't to do with 'passing off'. (It's highly unlikely that anyone could assume that a local chippy was associated with an international toy company). It was because 'R Us' was part of their registered trade mark.