Why would you add an unnecessary relay when you could just run through the E-Stop and be done with it?.
The safety relay has dual contacts, and dual monitoring. It sees both contacts close, and monitors that both contacts act the same way. If they don't within a few mS the relay faults. Relays use the concept of redundancy. A contact failing has (fake info) 1-in 2,000,000 chance of failing. The chance of two contacts failing at the same time is extremely tiny.
The reset on a safety relay automatically stops you from holding down the reset button to make the line run as well. I think I've seen some that will still function that way though.
Manufacturers have created these devices with these features for us. We shouldn't be taking estops straight to STOs in this day and age. They spent a lot of money testing, and designing that device with approved usages in their manuals.
https://literature.rockwellautomation.com/idc/groups/literature/documents/um/440r-um013_-en-p.pdf
page 45 starts showing examples for applications. All of them use the safety relay to fire a safety contactor. The safety contactor contacts should go to the STO.
In our risk assessment, we would test the circuit thoroughly and document it. This would involve me and my electrician going to every safety device, taking one wire off of each dual contact and testing. We would also have every safety device pulled into sistema, which helps cross reference all of the devices you are using in the safety circuit, and what your maximum safety rating can be with those devices. Its free.
https://www.dguv.de/ifa/praxishilfen/practical-solutions-machine-safety/software-sistema/index.jsp
ISO 12100:2010 specifies basic terminology, principles and a methodology for achieving safety in the design of machinery. It specifies principles of risk assessment and risk reduction to help designers in achieving this objective. These principles are based on knowledge and experience of the design, use, incidents, accidents and risks associated with machinery. Procedures are described for identifying hazards and estimating and evaluating risks during relevant phases of the machine life cycle, and for the elimination of hazards or sufficient risk reduction. Guidance is given on the documentation and verification of the risk assessment and risk reduction process. This costs around $220.
https://www.iso.org/standard/51528.html
ISO 14121-1:2007 establishes general principles intended to be used to meet the risk reduction objectives established in ISO 12100-1:2003, Clause 5. These principles of risk assessment bring together knowledge and experience of the design, use, incidents, accidents and harm related to machinery in order to assess the risks posed during the relevant phases of the life cycle of a machine. This costs $97.
https://www.iso.org/standard/38479.html
With your current question that started this post. You shouldn't be designing safety alone. Hopefully you don't wing it, you go to your boss, and you ask for help. The above is to help educate you. It shall not encourage you to complete this task alone. Don't do it. Be Ethical when facing a deadline.