Smaller cogs on the rear would help. So would a larger front chainring.

The problem with replacing cogs is that your bike likely does not have a 'cassette' - it has a 'freewheel.' A freehub type hub has the freewheel mechanism built in and the cogs slide on. A freewheel hub has a set of cogs with a built-in freewheel mechanism threaded on, and they are a lot less common these days, and you might have trouble finding one with considerable smaller gears. Modern cassettes are regularily found down to 11 teeth, which would help you out a lot, but you would need to replace your rear wheel, which is more expense than it is worth... unless your wheel is already bent or the bearings are shot.

If your crankset has bolted-on chainrings (many are riveted on) then you should be able to find larger replacements. MTBs from that era usually had cranks with 110mm bolt circle pattern, and this size has become very common again as it is used in modern 'compact' road cranks. Compact road cranks usually have two chainrings - 34-50 is common - but you could likely leave your 28 and 38 alone and just replace the 48 with a 50 or 52 without too much trouble. If you are using the same cadence as you have been to reach 30mph with your current setup, you could reach ~32 and 33 mph with a 50 and 52 tooth, respectively. If you can find a freewheel with a 13 tooth cog, your top speed with a 52 tooth chainring would be 35.5 mph.

(all speeds calculated with Sheldon Brown's Gear Calculator)

One more thing - if you are strong enough to get your bike up to 30mph the next step is to bring up your cadence... "spin to win" is the old saying.