(This reply was written but not posted, before I saw CopyrightAid's reply - apologies for any overlap in our advice)
Your question raises a number of issues. First of all, in common with a number of other technologies, the development of the internet over the past couple of decades has raced ahead of the law's ability to reflect these developments. The Copyright, Designs and Patents Act (CDPA) was passed in 1988 when the internet was relatively undeveloped and the world wide web had still to be 'invented'. So what we have today is a combination of the basic CDPA, caselaw (ie judge's decisions on specific cases) and a series of amendments as a result of European Union Directives or similar outside pressiures, brought into law by secondary legislation. But even still the law lags behind in some areas.
Turning to your specific question, a website is generally regarded as a literary work, in that it is basically made up of words (eg HTML or scripting), and software which is also specifically protected in most cases through the 1992 Copyright (Computer Programmes) Regulations. But in order to gain copyright, in common with other sorts of work, a website must be 'original'. This does doesn't mean 'unique', just that its fundamental structure and content must not be copied and it must be the product of the “sweat of the brow” of its creator, not merely a case of a monkey let loose with a copy of Dreamweaver. In other words, most websites will meet this standard.
The CDPA says that person who creates a literary work (the author) is the first owner of the copyright unless he is employed by someone else for that purpose. Here employed means a full employee, not a contract worker or a freelance who is commissioned for a specific job. The law on this was clarified by the Court of Appeal recently during an unfair dismissal case in which a contractor who had worked within the same company for some years and who was fully integrated into the company's management structure for line management purposes was nonetheless judged to be a contractor not an employee.
So on that basis, in the absence of any written agreement or contract with the company to the contrary, you would own the copyright to that part of the website which you created. The fact that the company has provided a brief for what it wants may have a bearing on whether you own the copyright outright or have to share it with the company. If they provided the 'idea' and you are merely turning that into a working site, it is probable that this would be a work of joint authorship, which means you would both own all of the copyright. If the two elements which you contribute can be separated (as for instance where a lyricist and composer write a hit song) then you can each own copyright in you own part, although on the face of it I am not sure how this could be done in the case you have outlined.
Ideally there should be an agreement drawn up which specifies exactly who will own what. This could cover your right to develop the standalone element you mention, independently of the company. If the copyright is jointly held, it is difficult for one or other of the parties to exploit part of the work without paying the other party a licence fee. The agreement can include other matters as well, such as any ongoing site maintenance arrangements. You should make sure that you also get a suitable credit as the author of the site (your moral right to be credited, which runs alongside the copyright)
You can find a specimen agreement which might be suitable here
http://www.own-it.org/contracts/57 although if the company are big and wealthy enough, they may choose to get their lawyers to draw up a bespoke agreement.