Can I add a static route Cisco3750Stack#ip route 0.0.0.0 0.0.0.0 10.6.0.1 10 to the existing static route and be able to test connectivity with the new IP scheme

No you can't test in the way your described because your testing static route won't be active until and unless exist route is in disabled state or deleted  . Basically layer3 device gives high priority to lower AD value route entry second priority is given to second lowest AD value route entry .. so in your case if existing route is having least AD value and newly configured route with AD value 10 in this case all the traffic will choose existing router entry with lowest AD value  .

Two test your requirement..two options can be considered

1. Take the downtime and delete existing route entry .so that newly configured route entry will be activite traffic forwarded based on this route

2. configure static floating route entries with similar Administrative value (AD) . So that traffic is equally load balances among two route and your can check accordinglly whether your second route is working as per your requirement or not

Can I add a static route Cisco3750Stack#ip route 0.0.0.0 0.0.0.0 10.6.0.1 10 to the existing static route and be able to test connectivity with the new IP scheme

No, you can't test in the way you described because your testing static route won't be active until and unless the existing route is disabled or deleted. Basically, a layer 3 device gives high priority to lower AD value route entries, and second priority is given to the next lowest AD value route entry. In your case, if the existing route has the lowest AD value and the newly configured route has an AD value of 10, all of the traffic will choose the existing route entry with lowest AD value.

To test your requirement, two options can be considered:

1. Take the downtime and delete existing route entry, so that the newly configured route entry will be active and traffic will be forwarded based on this route.

2. Configure static floating route entries with similar Administrative Distance values (AD). Traffic will be equally load balanced among the two routes, and you can verify that the second route is correctly working.