All GPS receivers provide ellipsoid heights; they do NOT provide elevations.
The difference between zero elevation on the North American Datum of 1988 (NAVD88) and zero ellipsoid height for South Louisiana is about twenty seven meters = 88.5 feet. Handheld GPS receivers (consumer grade), receive single frequency Coarse Acquisition code (CA code), where with Selective Availability turned off, has a nominal positional accuracy under good conditions of about ±15 feet in the horizontal.
Since the vertical accuracy of Handheld GPS receivers is about seven times worse than horizontal positions, 15 × 7 = 105 feet + 88.5 feet = ±194 feet vertical accuracy under good conditions!
That’s not bad for a $99 GPS receiver. For about $12,000 one can purchase a dual-frequency geodetic-quality GPS receiver that can provide ±2 centimeter accuracy over two days’ observations of at least 30 minutes each day, depending on baseline length. Of course, post-processing will require additional computer time to reduce the observed ellipsoid height to an elevation referenced to NAVD88. That is not possible with a handheld GPS receiver.