I have found couple of solutions:
1. I have created my own base map (tile package) for the area and replaced it with ESRI Base map. That allowed me to zoom to the level what I set during my tile package creation. The process is written here:
43247 - Set a custom scale range for a tile service in ArcGIS Online
The main problem in this process is area coverage can not be bigger. Then you would end up creating gigs of file size which is difficult to upload. I created separate base maps for small areas.
2. I found some free web services made available by state government to use as base map to ArcGIS Online. I have taken one of that and replaced the ESRI base map with it. This option is working better for me as I do not need to create separate basemap for different area any more. One is working for whole state. The web service I am using is here:
Access LPI Web Services - Land and Property Information