If unexpectedly in IMC, does it help to climb into clouds to get on radar? I gather that in the flatlands, probably not. But here is a tragic accident where a VFR-only pilot attempted to navigate steep mountain passes in Utah (Evanston, Wyoming west to Coalville, Utah, and south on to Heber City).  The clouds were reported to be closing in, but he wanted to go see for himself, and  promised ATC he would quit and go back "if" (when) visibility got too bad.  He was too successful following the freeways, and got cocky. Made his turnaround decision too late, turned around successfully, and headed back out the way he came - but collided with terrain nonetheless.  Reports compared with maps make it sound like he got turned around alright using the sky over a reservoir, and my guess is he lost contact with his "concrete compass".  It was terrible judgment all around.  Shouldn't have been there. 

He did have a squawk, and was periodically picked up on radar.  He was too low for radio without relays. 

Would it have been a viable strategy, instead of staying low in known mountains hoping for a sliver of navigable air between clouds and rocks... to boldly climb into the clouds, get seen by ATC, get enough altitude for radio, apologize and ask to be vectored back to usable VFR territory?
Absolutely not! If a pilot is not trained for instrument flight it would be near suicide to attempt flight in cloud due to spatial disorientation. In clouds you have to trust your instruments solely and be able to interpret them, which requires special training, retraining and practice. Without these pilots do not last long in cloud, see this related question for lots more details on inadvertent flight into IMC, which is applicable to your question. 

What the pilot should have done is made a precautionary landing on one of his concrete compass lines or a field while he still had the option to do so. Instead, he had get-there-itis and got himself perished as a result.