The hike up Beacon Hill can be done any time of year. During the winter I usually carry microspikes along just-in-case. It is a genuine multi-use area that is shared by mountain bikers, hikers, paint ballers and rock climbers (lots of nice granite).
The lower section of the hill is managed by Spokane County Parks with much of the remainder being private property. All of it is being managed for public use and much work has been done to keeping access open and on developing a plan for a formal trail system.
There are so many variations to reaching the top that even on a busy summer day, it's not uncommon to seemingly have the place to your self. Pick one that goes up and it's hard to go wrong. Stretch it out further by descending from the top towards Esmeralda Golf Course and finding a different trail to return on.
From John Shields Park, most trails will reach the top in about a mile. The trails are a mixture of double and single track that stay in the trees until the last 100 feet. Wildflowers are plentiful, especially balsamroot.
There are several access points for a hike up Beacon Hill. Here are the directions for getting to the easiest access, which is located at John Shields County Park: Take the Argonne exit 287 off of I-90 and head north on Argonne Street. Cross the Spokane River at 1.1 mile and stay in the left lane. At 1.6 miles, you'll reach Upriver Drive where you'll take a left turn. At 3.9 miles, you'll see a parking area on the north (right) side of Upriver Drive (not signed). Turn into the parking area, which is John Shields County Park. If you miss the turn, the next feature you'll see is a dam on the Spokane River.
ResourcesGoogle Maps: Beacon Hill Summit
Lots of excellent info including history of the area, plans for the future and maps:
Spokane County Parks