A linear regulator will reduce the noise but it may not be a large reduction at the switching regulator noise frequency. You need to look at the linear regulator's data sheet and see what attenuation it has to the switching frequency. It's typically much less than at 60Hz, for example.
Many good power supplies have a linear stage after the switching stage for better output characteristics like regulation. The noise could be attenuated if the linear stage is fast enough to react to the noise signal, but it wont help much if it's not.
One simple way to reduce output noise from the switching frequency is to use a small coil in series with the output and a parallel output capacitor of significant value. It's really amazing how much difference this simple filter can make especially if the switching frequency is 100kHz or greater, but not bad even if it's lower.
The idea is to make a small hand wound coil with several turns of wire that can handle the output current and use a nice large say 100uf cap in parallel with a 0.1uf ceramic on the new output. You can buy a coil if you like, but even an air core hand wound one with say 10 turns will work to some degree.
Of course it's a good idea to check the output to make sure the noise is down low enough for your intended purpose so you might want to use a scope to check it.
It's interesting that a while back data sheets for switchers didnt show any extra output filtering i think because they didnt want to make the designer think it would be a large parts count design, but recently i've seen more data sheets showing these kinds of extra filters on the output so the trend is changing now. It may be because of the significant difference even small value coils make in reducing switching noise.
Most linear regulators have limited high frequency power supply noise rejection. You need to passively filter the high frequency noise prior to linear regulator.
What you have to do is a quick analysis on how much output current you are going to draw and how fast it has to respond. The coil will limit how fast the power supply can react to step changes in load so your inductor and cap combination really should be analyzed in a simulation. For one test you would apply the load and see if the response is fast enough for your needs. It should be at least as fast as the linear that follows, unless you dont need it that fast.
Probably 2uH is a good starting point as ronsimpson suggested with a nice big cap say 100uf and small ceramic too. It also depends on the switching frequency as higher frequency spikes will be attenuated much easier than lower frequency spikes.
One mistake not to make is to move the feedback line to the *new* output. That could cause oscillation.
Can you check this with a digital scope? It depends on how good your scope is...can it catch a fast spike with good enough resolution? Personally i prefer analog scopes for these kinds of applications.