Alpha under .7 in experience sampling data for some measurement moments

I am facing a problem with my experience sampling data:

Some of my alpha (Cronbach) are under .70 for 25 out of 240 measurement moments. Sometime it is when there are less then 20 subjects but sometime not...

I am looking for literature to make a decision: delete those measurement moments (reducing my sample) or just work with a mean of the alpha in order to justify the reliability of the scale.

Does someone ever faced this situation and if so, what is a suitable solution?