Aye, you could spin with a broom handle and curtain rings, but might not be the best experience.
Loads of spinning rods out there, personally i'd go for one of the none specialist ones as they tend to be cheaper (on the whole). I bought a shakespeare odessa flattie for the purpose and it is absolutely excellent - but there are ofc other options (dont see the point of getting some of the top end flattie/bass rods from the rod making royalty though).
Few reasons for getting a more suitable rod.
You can thrash out 1-3oz rigs with ease
Much more fun/excitement when actually playing/landing the fish.
Very light - can use it all day
You can use the right rod for other light tasks - close in ledgering, float, feathering for mackeral.
I have an odessa flattie rod. As Cromcruaich said this rod will do everything. Float/spinning/feathers etc Put a largish fillet of mackeral on it or a live sandeel and try to get yourself a good bass. Hold on though......
If cash is tight, you would be better off buying somthing second hand or even a cheap 7' spinning rod that can be found for under £10 rather than use your pier rod which is not going to give you the distance you need with a spinner. Another alternative is an old carp rod.
Also important is the line/reel you are using I would suggest 12lb max. If you have a large reel, don't re fill the whole thing. Strip off enough line to put on 150-200m of the lighter stuff.
I have a a 20 year old course rod bought in france bought for 10 francs and a £6 pound fixed spool reel my line varies between 8 - 10lbs. I have added a lot of links that might help you choose some thing as a starter kit. the problem starts when you have to choose spinners. as there are so many styles and types to choose from for all sort of fish. if you know what type of fish you would like to catch any good tackle dealer will give you information on the best type of spinner/plug/lure for you to use. Once you have an idea about what you would like to spin for ask a question about it and how to go about it. also if you find an outfit that will suit your needs ask a question as someone may have it all ready and they can tell you what they think about it before you spend your hard earned money on it.