I would actually pipe to base search rather than additional where clause for two reasons:
1) Filtering records upfront in base search is faster.
2) search fieldName=value is faster than where fieldname=value
index=<your_index> date_hour>=6 and date_hour<=12
date_* default fields are not available for all sources, for instance they are not present for Windows event logs.
You can try creating the field like below and then filter -
| eval date_hour=strftime(_time,"%H") | where date_hour>=6 and date_hour<=12