Flight Instructor reported a NMAC while in cruise flight that required evasive action to avoid a collision.

Narrative

I was doing an instrument flight with a student over the BYP VOR. Making frequent radio calls and scanning for traffic; supplementing with onboard ADSB-In capable equipment. While practicing holds; going west on the 090 radial; I scanned to the north and saw an aircraft same altitude. I immediately took control and made a climbing left turn; inquired on the local CTAF about nearby traffic once again; and heard nothing; I had no traffic alerts nor traffic on either ADS-B In equipment; as I looked back outside the aircraft had also begun their own evasive maneuver as they had gotten too close. I again called on the frequency and got no response. Although radios nor ADS-B is required in Echo airspace; utilizing these tools drastically increases situational awareness. I read Aircraft Y as the aircraft then popped up on ADS-B once they were about 2 miles south of BYP. After further research; the aircraft was traveling east cruising at 4500MSL which would not coincide with 91.159. Unfortunately; we both elected to climb. I am thankful that I was not heads down teaching an instrument student who was under a view-limiting device at the time. This goes to show the importance of effective scanning and not getting complacent. We both practiced see and avoid.