Take a hike up to Pinnacle Saddle in the Tatoosh Range. The trailhead is at Reflection Lakes right near Paradise. The final slope up to the saddle might be a bit of an avalanche hazard, so check conditions. There are also hikes out of Paradise, such as to Panorama Point, or up to the Paradise Glacier. All of those hikes are low-key, but are sure to have a lot of snow on them this time of year. An ice ax will definitely be necessary on Pinnacle Saddle.

Stunning views can be had from the "Skyline Trail", trailheaded in Paradise, although you'll see most of those views from your climb.

If you want acclimitization, do the Camp Muir hike. By June, the snow should be firm and the boot track should be solid as long as there are no more big snow dumps. Of course, don't go if the weather is suspect and always take the 10 essentials including the park service "Get Your Bearings" printout showing compass bearings to and from muir in case of a sudden whiteout. On a good weather day, you won't be alone.

If you want to get out of the park and have 4WD with good ground clearance available, go down FR52 to FR84 to FR8440 from ashford and do the "High Rock Fire Lookout" trail. It's a relatively short ridge hike up to a prominent pinnacle overlooking most of the forest south of Rainier. Again, stunning views with an airy feel. Before you go to this one, check the USFS site for road conditions in Gifford Pinchot National Forest, roads 84 and 8440: http://www.fs.fed.us/gpnf/recreation/cu ... oads.shtml to make sure you can get up. It's highly possible that the road will be open before the page says it is. It's also highly possible there will have been a landslide or tree across the road blocking progress. Trailhead is at Toehead Gap, about 4300ft. http://maps.google.com/maps?f=d&source= ... 6&t=p&z=15

all these will be fine without crampons and with/without plastics or hiking boots in good weather.