You can always retrieve more information from system views or dmv's as below. Whatever information you will want to pull out by testing local session and whatever is available, then you will change the table structure that will hold the data and the insert statement in the trigger if that's what you are looking for.

-- older version but still supported

SELECT [SPID], [kpid], [blocked], [waittype], [waittime], [lastwaittype], [waitresource], [dbid], [uid], [cpu], [physical_io], [memusage], [login_time], [last_batch], [ecid], [open_tran], [status], [sid], [hostname], [program_name], [hostprocess], [cmd], [nt_domain], [nt_username], [net_address], [net_library], [loginame], [CONTEXT_INFO], [sql_handle], [stmt_start], [stmt_end], [request_id]

FROM master.sys.sysprocesses (NOLOCK) WHERE SPID = @@SPID

-- newer versions

SELECT [session_id], [most_recent_session_id], [connect_time], [net_transport], [protocol_type], [protocol_version], [endpoint_id], [encrypt_option], [auth_scheme], [node_affinity], [num_reads], [num_writes], [last_read], [last_write], [net_packet_size], [client_net_address], [client_tcp_port], [local_net_address], [local_tcp_port], [connection_id], [parent_connection_id], [most_recent_sql_handle] FROM sys.dm_exec_connections (NOLOCK) WHERE session_id = @@SPID

SELECT [session_id], [login_time], [HOST_NAME], [program_name], [host_process_id], [client_version], [client_interface_name], [security_id], [login_name], [nt_domain], [nt_user_name], [status], [CONTEXT_INFO], [cpu_time], [memory_usage], [total_scheduled_time], [total_elapsed_time], [endpoint_id], [last_request_start_time], [last_request_end_time], [READS], [writes], [logical_reads], [is_user_process], [text_size], [LANGUAGE], [date_format], [date_first], [QUOTED_IDENTIFIER], [ARITHABORT], [ANSI_NULL_DFLT_ON], [ANSI_DEFAULTS], [ANSI_WARNINGS], [ANSI_PADDING], [ANSI_NULLS], [CONCAT_NULL_YIELDS_NULL], [transaction_isolation_level], [LOCK_TIMEOUT], [DEADLOCK_PRIORITY], [row_count], [prev_error], [original_security_id], [original_login_name], [last_successful_logon], [last_unsuccessful_logon], [unsuccessful_logons], [group_id] FROM sys.dm_exec_sessions (NOLOCK) WHERE session_id = @@SPID

SELECT [session_id], [request_id], [start_time], [status], [command], [sql_handle], [statement_start_offset], [statement_end_offset], [plan_handle], [database_id], [USER_ID], [connection_id], [blocking_session_id], [wait_type], [wait_time], [last_wait_type], [wait_resource], [open_transaction_count], [open_resultset_count], [transaction_id], [CONTEXT_INFO], [percent_complete], [estimated_completion_time], [cpu_time], [total_elapsed_time], [scheduler_id], [task_address], [READS], [writes], [logical_reads], [text_size], [LANGUAGE], [date_format], [date_first], [QUOTED_IDENTIFIER], [ARITHABORT], [ANSI_NULL_DFLT_ON], [ANSI_DEFAULTS], [ANSI_WARNINGS], [ANSI_PADDING], [ANSI_NULLS], [CONCAT_NULL_YIELDS_NULL], [transaction_isolation_level], [LOCK_TIMEOUT], [DEADLOCK_PRIORITY], [row_count], [prev_error], [nest_level], [granted_query_memory], [executing_managed_code], [group_id], [query_hash], [query_plan_hash] FROM sys.dm_exec_requests (NOLOCK)

WHERE session_id = @@SPID