IANAL but there usually isn't a problem.
To add to leahzero's comments, there is a legal aspect which helps you.
For example, imagine that you asked me for a job reference. I write a glowing report and email it to you. You attached that glowing report to your resume and mailed it to a hundred people in your search for work.
If I had the foresight to register the copyright my glowing report of you - could I sue you for copyright infringement and make a minimum $600 for each violation - a total of $60k? (I'm assuming USA law for the sake of simplicity)
After all, technically I'm 100% in the right - I hold copyright in that piece of writing. You violated copyright by copying it.
Thankfully, however, there's an escape clause for the 'technically you violated copyright' argument.
Because you asked me for a reference, unless I specifically said "Don't copy this", I gave you an 'implied license' to use it in the way that a normal job reference is used.
This has been tested in the courts. Believe it or not, there was a graphics designer who created a logo for a company, then sued the company for putting the logo on their business cards!
The contract didn't say "Work for hire" and so when the company decided to use another graphics designer for the next job, the designer figured he could sue the company - or at least force them to take his logo off their business cards.
It didn't work. The court decided that while the designer still owned the copyright (since it wasn't work for hire) the 'I'll design a logo for you' offer implied that they would be able to use the logo for all the normal things logos are used for.
So, unless otherwise stated, you can argue that there is a general understanding that you'll use snippets of the footage that you shot in your director's reel.
It's an 'implied license'. That's the legalese version of what leahzero was saying.
On the other hand, this is actually a great opportunity for you.
You can use this as an excuse to give a followup call to all those people you worked with on past projects.
And if those people happen to mention some new work available .. then you win as well !
Mac
(PS: Definitely check that you are fine under non-disclosure agreements though)