Sensor rise time compared to experimental step input rise time
So, when you remember how the step response, impulse response and frequency response are related, you'll notice that if you, instead of an actual impulse integral (i.e. a step) use something that is wider, then the frequency response simply gets windowed to lower frequencies.

In other words, and to put it as an information-gathering problem:

if the bandwidth of your excitation signal is limited (e.g. by a slow rise time of your step), then the information you can get about a system reacting to that is limited to the frequencies in the excitation.

It's often physically hard to make a large jump excitation. But maybe smaller jumps work, and you'd be better of


Generating a pseudorandom (band-limited) white signal (and save it!)
exciting your system with that
correlating the output of your system with the known white signal


The idea is this: you get the frequency response as the transform of the impulse response.

A white signal $s$ has the property ($*$ representing convolution here, ${}^*$ meaning complex conjugation, which is identity for real-valued signals)

$$s(t) * s^*(-t) = \delta\text,$$

i.e. you can "generate" an impulse $\delta$ by correlating a white signal with itself. 

Knowing that your system has a step response and impulse response $h$, and knowing that convolution is commutative, we can do the following to find that:

$$\delta * h = s(t)*s^*(-t) *h = s(t)*h*s^*(-t)$$

In other words: instead of exciting your system with an impulse, which can be hard to do, to get the impulse response, we excite it with a white signal, which we can (withing the bounds of our time and bandwidth restrictions) produce digitally. 

Then, we just correlate the output of the system with the same white signal, and get a result that is the same as had we put on an impulse of the same bandwidth as the white signal – but the white signal can be made arbitrarily long, so that very small amplitudes of excitation suffice to deliver the same energy as a really high, short pulse. 

This is something that is commonly done to identify systems (especially: communication channels) in digital communications, but it applies to any LTI system.