For the above links, remember to include why your moving from part-time work to full-time work can benefit your company.
Reasons may include:
1) Increased productivity for your company.
2) Proof that your previous performance has already benefited the company financially as well as in terms of reputation.
3) Financial savings for your company through your ability to do more and varied tasks. Some of these tasks may have previously been hired out to private contractors, an expense for your company. You could use this argument if you have just completed a degree (undergraduate or graduate) which adds to your skills.
There are, of course, more reasons you may include. Here is a pertinent article:
In order to craft a letter to go from part-time to full-time work, a few basic factors need to be considered:
1) Is the company where you work openly advertising full-time positions or are these unadvertised positions?
Both advertised and unadvertised full-time positions may involve different procedures in the application process. Also, some managers or heads of departments may be given wide latitude in how they will hire for unadvertised positions within a company. Either way, careful research will arm you with information on the best way to proceed.
For example, your contributions may be well known to the hiring manager and/or department head. If so, your letter should not only include all your accomplishments within the company, but how these contributions will increase the department's effectiveness and profitability margin. Your offer may prove more attractive if the hiring manager can see that hiring you will add to his/her position and reputation as an effective administrator.
2) Are these positions open to both employees of the company as well as members of the public?
Again, there may be specific procedures for applying for these types of positions. Adequate research will ensure that your application is thorough and effective.
3) Have there been recent resignations at the company or have positions opened up as a result of employees having left?
Whether someone has left a company for another position or for personal reasons, effective communication with your hiring manager regarding these open positions often require great tact and finesse (before you even start the application process).
With all that said, here are links to cover letters for advertised as well as unadvertised positions.
Cover letter for an advertised job.
Cover letter for an unadvertised job.