the ssd will use the power it needs. the psu only gives what is asked of it. it does not determine what is given. an ssd uses a lot less power then a spinning hdd. around 4-5w usually.
if your psu is old and very cheap it is worth replacing it. tons of good prices right now with the black friday sales. the psu is the engine of the system. you'd not buy a lambo and then ask for a cheap 1.0 ltr 3 cylinder engine would you to save a few bucks?
we can easily offer some options if you are interested. i've seen a bunch of good prices on various models and power levels